Then how did Uriel Septim bear the Amulet of kings without the soul of a dovah, and what about Martin Septim's sacrifice to take the form of a dragon to defeat Mehrunes Dagon, how would this be possible without being a Dragonborn? This is simple to explain, its because they were in fact Dragonborn.
Possible, seeing as talos is now a god and that could have given the entire Bloodline to be blessed to be born with the soul of a dragon, but there is no proof either way. And still doesnt mean its hereditary. The septim line came from a man-now god, that could have something to do with it. And seeing the very few amounts of dragonborns in history, there would be a lot more if it was hereditary.