Well, repeating what Gandalf said, but comparing to the two strongest DPR feats we have, which are Polearm Master and GWM. Also, with math.
The feat becomes really strong - not completely broken though.
Without the feat, you would be doing 2d6 + Modifier, which at low levels (i.e. \$<4\$) means 2d6 + 3. With the feat, this increases to 2d8 + 6, which is a +5 damage on hit bonus, or \$+5 \cdot P\$, where \$P\$ is the probability to hit, increase on your average damage.
For reference, let's compare it to the other mentioned feats. Let's say you are fighting against an enemy with AC 13, which is the default value for low levels and a quite common number. Assuming you are using a Greatsword with Great Weapon Fighting (optimal damage), GWM increases your average damage by around 1.6 at no cost. Polearm Master increases your average damage by 2.2 - at the cost of a bonus action.
Your feat increases 2.85 at no cost. You could think "Oh but you use the bonus action" - but actually, you would already be using the bonus action if you were fighting with two weapons. The feat itself has no cost.
Against some other creatures, like a Zombie with 8 AC, your feat still gives a +4, while GWM gives +4.4 and Polearm Master a +3.2. Note that, again, Polearm Master has a cost, and GWM is specifically good against lower AC enemies. Against an enemy with Splint (17 AC), your feat gives +1.85, polearm +1.44 and it's better to not use GWM. Point being: your feat became competitive with the strongest DPR feats we have for melee, arguably outclassing both due to having no cost and being more versatile since it doesn't suck against high AC enemies. Sure, polearm master gives you reach and the better Opportunity Attack, but the feat provided a greater DPR increase than Polearm Master.
The reason I say it's not completely broken is just because the DPR increase will fade away as levels go up and extra attacks become more common, magic items become common but not enough for you to have two +X weapons probably, while GWM will just get stronger, and Polearm Master will probably be combo'd with Sentinel or a Cavalier archetype for a Defender-style play, or actually Polearm Master will be used WITH GWM.
But now my major point:
The fighting style becomes useless.
Both Polearm Master and GWM usually go with GWF for damage optimization. With your feat, you could dual wield without actually taking Two Weapon Fighting. This is the major problem. You would probably take defense and get a free +1 AC for that. Usually, homebrewing something that completely overlaps and overshadows another feature - in this case, the TWF fighting style - is bad design. You could change TWF in order to make it useful again, or you could stack the modifier bonus (please don't), but that's probably going too far.