I'm A Banana
Potassium-Rich
2d6 damage is fairly potent...for those levels where you and your allies are one-hit-wonders that can straight-up die to a bugbear's crit. But 4d6 damage at level 11, when Fighters are getting three attacks (3d8+Str/Dex is substantially higher than 4d6, and that's before counting crits), Paladins get improved smiting, etc.? Kinda weak [edit: by comparison], actually.
Apples : Oranges. A racial feature doesn't need to be equal in power to a class feature (especially one at a higher level). A player doesn't need to pick between a dragonborn and a fighter and a paladin. A character can be a dragonborn AND a fighter or a paladin. Or, heck, a dragonborn fighter/paladin for all that.
But if you are going to compare them, 4d6 in an area is actually pretty comparable to 3d8+str/dex, if not a little better in places where you can snag multiple wee beasties at once.
Where's the fluffy stuff, the bonuses to history or intimidate or a minor healing benefit?
"I'm a frickin' dragon-person" isn't enough?

It probably wouldn't break the game to give 'em something like Advantage on Charisma checks when interacting with dragons or somesuch. It's niche enough.