No, but there is proof that it is bestowed. Someone have to get it from Akatosh, even if it happens to be herditary. Before Bethesda clearify this I am not going to change my mind, because I do not see any proof that it is hereditary.
The Nerevarine of course was a Dragon-born as well... As was the current Dragonborn of Skyrim. Considering that the hwole premise of the TES games is that we can decide where our character came from, I think that pretty much disproves that is hereditary.
Also:
"It is unknown how Mankar Camoran was able to wear the Amulet of Kings, since only those of Dragon blood can wear it. However, it is said in the Making of Oblivion Documentary, that Mankar Camoran is descended from another line of Princes, and is trying to claim Tamriel for them. This may provide insight as why he can wear the Amulet of Kings."
As for the Septims, with the exception of Tiber Septim, there is NO evidence to suggest that any of them were Dragonborn, just that they had the "dragonblood" - i.e. they were missing the dragonsoul required to be dragonborn.
"But should these Dragonfires fail, and should no heir of our joined blood wear the Amulet of Kings, then shall the Empire descend into darkness...."~ Trials of St. Alessia, also found in Imperial City loading screens
It seems to me that the dragonfire and dragonblood protection of the Empire does not indicate that the Septim empires are dragonborn..
There IS NO EVIDENCE TO SUGGEST THAT a child of a dragonborn was a dragonborn
Fixed for you.