Being Dragonborn isn't a trait passed down through the generations(Well, it technically it could be. Septim Bloodline and all that).
But Ultimately, it was a gift bestowed upon Man/Mer by Akatosh. So Yes, an Altmer could be given the gift if favored by Akatosh.
Don't find anything about svcking the souls of dragons to be realistic.
But yes, what Eniroth said is right, it's a gift that's bestowed by Akatosh, race does not matter, he could have given the gift to a mudcrab if he wanted too.
Mankar Cammoran (Oblivion) was drangonborn, so why not?
Do gods need to act in a realistic manner, or abide by mortal concepts of propriety. If Akatosh wants a Scrib to be a Dragonborn, you get a kwama Dovahkiin as well. It's not as if Elder Scrolls gods are the most sane or logical entities.