I'm sure I did an exploration of this when the 2024 PHB came out two years ago, but I thought I'd do a revisit.
When you're a weapon-wielder in a fantasy RPG, there are basically three ways you typically go: you either have a weapon in one hand and a shield in the other, a big two-handed weapon, or you wield a weapon in both hands. (There was a paired set of shields in Dark Souls III that I always wanted to build a character around, though I think you couldn't even get it until you made it into the Ringed City DLC).
It's pretty obvious that the intent behind a "sword and board" build is that you're sacrificing a bit of damage in the name of survivability (though depending on the game's mechanics, this need not always be the case). But both the dual-wielding and the "great weapon" style (we'll call it that because that's the general language used for it in D&D) are oriented toward doing lots of damage. Generally, the way they work is that the two-handed weapon is meant to hit very hard with each swing, whereas you make up for the lightness of dual-wielded weapons by making more attacks.
This is very true in D&D, and is, I think, even more true in 5.5.
Broadly speaking, I think that there's a feat that's designed for each playstyle, though there are others that might come into play (Polearm Master works great with two-handed reach weapons, though it can also work quite well with a spear-and-board or quarterstaff-and-board build).
Great Weapon Master was significantly changed in 5.5, but I'd argue it's more powerful now. I initially did a bit of math to compare it to the 5.0 version and found that even in tier 1, when the bonus damage is only +2, it's going to do a bit better than the old version.
(Quick and dirty version: if you've got a +5 to hit with your weapon, attacking a Goblin Warrior with the old version gives us +0 to hit and we deal 2d6+13 on a hit. With an AC of 15, that means we have to roll a 15 or higher, or a 30% chance to hit. So, our 20 average damage is only 6. With the new version, we need to roll a 10 or higher, so we've got a 55% hit chance. Our damage is only 2d6+5, or 12 on average, but that means 6.6 average damage, so we're ahead. Crits aren't really affected by this. Against a Zombie, to be fair, it's 65% of 20, or 13, versus 80% of 12, or 9.6, so it's not a always better.)
Still, the Great Weapon Master feat is an obvious go-to for players who want to wield a Heavy weapon. Indeed, it's actually recommended even if you're a ranged weapon user, as this will buff Heavy Crossbows and Longbows (it feels like a Hunting Rifle should also be a heavy weapon, and maybe a Musket. Let firearms be viable in D&D!)
Now, there's also a fighting style for Great Weapons. The bad news is that it actually sucks. Even its best-case scenario, a 2d6 weapon, winds up getting only a very slight increase in damage, going from 7 to 8 average damage.
By contrast, the Two Weapon Fighting style actually adds a fair amount of damage each turn, letting your bonus attack from the Light property to add your ability modifier. Not only does this mean potentially 5 more damage per turn (which someone with the Great Weapon Fighting style would need to make 5 attacks to get from its fighting style feat, which is going to require something like an Action Surge) but it also just makes the character easier to run because you don't need to remember not to add your ability modifier to the damage...*
Then, the corresponding General Feat is, appropriately enough, Dual Wielder. The 5.0 version of this allowed you to dual-wield non-light one-handed weapons, such as Rapiers, Longswords, or the like. It also let you draw and sheathe them together (which is kind of now just a base ability in the game) and would give you +1 to AC while dual-wielding (half-making up for not having a shield).
In practice, though, this only winds up raising the damage of the weapons by 1 each, as you go from d6 weapons like Scimitars and Handaxes to d8 weapons, the highest value for a one-handed weapon (with the exception of the 5.0 lance, though only when you were mounted).
The new version of the feat looks really weird when you first read it - it allows you to make a bonus action attack without adding your ability modifier with any one-handed weapon, as long as you attacked this turn with a light weapon. If you wanted the pseudo-Titan's Grip (that's a WoW reference) effect, you might find that odd. But the point is not to let you wield heavier weapons. Instead, it comes down to weapon mastery, and specifically, the Nick property.
Nick allows you to make your off-hand bonus action attack as part of the attack action, preserving your bonus action. This is great if you need the bonus action for something like Smite, moving Hunter's Mark, or whatever.
But the Dual Wielder feat then allows you to take that unused bonus action and do what you were going to do with it anyway, which is to make another attack. This one doesn't add the ability modifier, so you will still need to remember that (if it did dual-wielding would be significantly more powerful).
Thus, I think the intent for most dual-wielders in 5.5 is that they will be making two more attacks than they would if they only had a single weapon.
Now, I don't know if this will always work. In 5.0, the three classes that got Fighting Styles had curated lists, and Paladins, for example, didn't get Two-Weapon Fighting. While they now have access to it, the change to Divine Smite has also made it a little less appealing, as we might want to save our bonus action in case we land a critical hit.
The math is very complicated, because many light weapons have the Vex property, so while you'll need a Nick weapon for your off-hand (the Scimitar generally being the optimal one, though a Dagger can also work, especially if you want to pair it with a Shortsword and take the Piercer feat,) and so the actual hit chance you get three attacks into a turn where previous hits are granting advantage is, well... you've seen me slog through that on this blog.
Still, I think generally when I've done all that math, it's pretty rare that a dual-wielding build actually outdamages one based on Great Weapon Master, regardless of class.
That being said, I don't generally build for pure optimization. I'm not doing all sorts of funky multiclass builds, and vibes are, to me, more important than being super-powered in D&D, because, well... the game's kind of balanced to let players win most of the time. I wouldn't make a, say, Paladin who had a +0 to Charisma (which one of my friends threatened to do when we were starting a campaign - now he's multiclassed into Warlock, so obviously he didn't) but "not doing the most possible damage" isn't the same as "doing terrible damage."
With all of that out of the way, then, how would we want to build a dual-wielder?
First off, let's talk classes that it works for. Obviously, you'll want to play a class with Weapon Mastery. This does mean it's not as appealing on a Bladelock (who also runs into the problem of only having one Pact Weapon at a time) or a Battle Smith Artificer. Valor Bards might go for Sword-and-Board instead.
But this means that Fighters, Rogues, Paladins, Barbarians, and Rangers are all decently situated to be dual-wielders.
I think the playstyle fits thematically well with Rogues and Rangers in particular. You can play a Strength-based Ranger (I did for the first 5 levels of Tomb of Annhilation, though the campaign fell apart shortly after we got to Omu) but both classes I think are intended more to be Dexterity-based. For Dex-focused characters (with the exception of heavy ranged weapon users,) Great Weapon Master isn't an option, so you're sort of forced to try something different anyway.
Fighters, Paladins, and Barbarians are all typically Strength-focused, but Strength-based dual-wielding is totally doable - the feat lets you use either Strength or Dex as a prerequisite and as the stat that gets boosted by it. This means that your options are slightly expanded, as you can use Handaxes, Light Hammers, and Sickles, which are all Light weapons that don't have Finesse.
I think where you might run into some friction is in how much you need your bonus action. Barbarians will need it at the start of combat to get into their Rage, but typically, once that is active, they'll have it up for the rest of the fight (and if you have multiple fights in quick succession, or burn a rage outside of combat before a fight, you might not have to pay that price). And then, getting to add their rage damage bonus to another attack is nice.
Fighters are actually among the least bonus-action-dependent classes, with mainly just Second Wind to use in an emergency. The only downside to dual-wielding as a Fighter is that they're already the class that gets the most attacks (well, Monks might disagree,) thus diluting how proportionately higher the damage overall is by adding attacks. That said, Fighters don't actually hit this point until level 11, so in tiers 1 and 2, it should be a pretty good benefit. Also, because Nick makes one of your off-hand attacks (and if we get really into the nitty gritty, technically by using the Nick attack as part of your action, this then lets you attack with your "main hand" weapon with the Dual Wielder feat bonus action attack) part of your Attack Action, that means you should be able to do it again when you Action Surge. So, by level 5, if you have the Two Weapon Fighting Style and a Nick weapon in the off hand, you can make 6 attacks in a single turn when you Action Surge. I think the fact that you can get so many attacks at higher levels does start to favor Great Weapon Master more, but prior to level 11, dual-wielding is going to be pretty sick on a Fighter.
Paladins, I think, are going to run into the most trouble, because they're actually quite bonus-action-happy, with Smite spells all taking a bonus action, but also with things like Lay on Hands now using a bonus action. The dual-wielder feat can still be good when you want to play conservatively (if your campaign is a bit more focused on slow dungeon crawls) but it might not be worth the feat if you're constantly skipping that attack to use Smite. All that said, you get a little more benefit from Radiant Strikes at level 11 if you're making more attacks, and while a single d8 more damage isn't going to be as good as the 8 damage you'd be getting from Great Weapon Master each turn, it does bring dual-wielding up a bit.
Rangers, of course, are classically depicted as dual-wielding when they're not using a ranged weapon. They are, to be fair, also pretty Bonus Action-dependent, thanks to Hunter's Mark, but at least as we've seen with the Hollow Warden, the folks at WotC are starting to give players some options that don't revolve as much around that spell (something I wish they'd realized when designing the 5.5 PHB - Hunter's Mark is a perfectly cool spell, but as long as it's preventing you from using any other concentration spells, it shouldn't be a central class feature). Honestly, outside of Hunter's Mark, they don't necessarily have a ton to do with their bonus action (except for the Beast Master and Drakewarden). I expect that if future Ranger subclasses follow the lead of the Hollow Warden, dual-wielding could very well be the default for Rangers (the Demon Hunter surely ought to be focused on dual-wielding).
Now, Rogues: Rogues do like to use their bonus action a lot. But they have sort of the opposite of the Fighter's issues, which is that they're the only martial class that doesn't get Extra Attack. Thus, that additional attack is worth that much more, proportionately. Furthermore, Sneak Attack does a lot more damage per turn if you have more chances to hit. To demonstrate: a level 8 Rogue has, say, +7 to hit and does 4d6 (14 average) damage with Sneak Attack. If they're attacking a target with an AC of 18, that's a 50% hit chance. So, the Sneak Attack damage is 7 average (ignoring crits). But if we attack twice, there's a 75% chance one of those hits, so it's now 10.5 average damage. If we attack three times, it's a 93.75% chance to hit, or 13.125 average damage. Yeah, there's diminishing returns, but getting that second attack pretty significantly increases the damage we can do, before we even get into the weapon's damage itself (Rogues don't get Fighting Styles innately, but it's maybe worth considering picking up Two Weapon Fighting at an ASI level even though it doesn't help us bump Dex).
Now, I think what I'm starting to realize about this is that, in a vacuum, just fighting with two light weapons can be quite good. In a world without General Feats, it might even be better than using a two-hander in some cases. I think the main thing that makes it seem less viable is just how powerful Great Weapon Master is (even if we're only paying attention to the added PB's worth of damage to each Attack Action attack). The Dual Wielder feat is a little finicky. It's the most obvious feat to pick up if you want to enhance the damage output of a dual-wielder, but it's also a feat that I don't think is quite as mandatory for a dual-wielding character as Great Weapon Master is for a character using Heavy weapons.
Ironically, this is sort of the opposite of the Fighting Style feats, where I think you'd be a fool not to take Two Weapon Fighting if you're dual-wielding, but I'd actually recommend not taking Great Weapon Fighting because its benefit is so marginal that you'd probably be better off getting some non-damage-oriented Fighting Style (Defensive's always great, though I also like Blind Fighting). To be clear, Dual Wielder is far better of a General Feat than Great Weapon Fighting is as a Fighting Style feat, but I do think there are some circumstances where you might just enjoy the benefits of the Nick property and call it a day. (I think I did that with by Oath of Noble Genies Paladin build, though that might have also gone for a Dex-based Sword-and-Board Dueling build).