TWF on a Barbarian. Sure, they don't get a fighting style, but here's what they do get:

1) Rage bonus damage on an additional attack every round

2) reckless attack adv. on an additional attack every round.

Let's do some comparisons. For this, we'll compare two shortswords (still using STR so we get rage bonus damage) to a great axe. We'll have STR values be the same, and appropriate for a Barb to have a various levels. We'll also be looking only at while-raging damage.

At lvl 1, the GA barb deals 1d12+5 damage each round (average: 11.5). The SS barb deals 2d6+7 (average: 14). Pretty large disparity for first level, on top of the fact that the Barbarian can split the damage up if they kill their initial enemy.

At lvl 5, Barbs get Extra Attack (and I assume they will pump STR at lvl 4), which levels things out a bit. GA barb deals 2d12+12 (average: 25). SS barb is sitting at 3d6+18 (average: 28.5).

Now check out 16th level. 20 STR by this time. GA barb has 2d12+18 (average: 31) while SS barb has 3d6+27 (average: 37.5)

So you can see that, even if you don't factor in accuracy, the SS barb actually deals more damage (and even though their criticals aren't quite as nasty, they have more chance to get them, which almost balances that out, making that difference minimal to say the least).

But then you factor in Reckless Attack. It's hard, if not impossible without bringing in very specific situations, to quantify the value of Reckless Attack, but it almost doubles your crit chance, on top of almost removing your chance to critically fail, and its also a majoy boost to accuracy. GA barbs get a maximum of 2 attacks per turn that can benefit from this, once of each of their Main Action attacks. But SS barbs get an extra use on all of their BNS attack attacks.

The only thing, though, is the need to rage. So the turn 1 damage of a GA barbarian is larger than the SS version, but once the second round hits, the SS starts doing more damage, so if the combat is going to last more than 2-3 rounds, SS will come out on top. You can even pick up a feat to improve your weapon dice to d8s, which gives you even more damage over GA, and that feat also supplies +1 AC and the ability to draw two weapons together, if your DM is really annoying about those types of things.