I've had this on my mind for a while now and am so boggled on this question. For all you people out there that are reading this and have played Oblivion know what I'm trying to ask. In Oblivion, the Septim Dynasty ruled Tamriel for hinderers of years, they also had Dragon blood in them which let them light the fire and all that other emperor stuff. In Skyrim how ever you are also Dragon born, obviously. So my point is, should the dragon born in Skyrim rightfully be the emperor in Temriel due to his dragon blood and his connection by blood with the Septim dynasty? Are you some long lost cousin of Uriel Septim or his heirs? Just something to think about and wonder.

Two different things.

We have the soul of a dragon. They have the dragon blood.

septim implies dragon blood

dovahkin has dragon blood

this does not allow us to say that dovahkin is a septim.

All Septims were dragonborn, not all dragonborn are Septims. Dragonborn does give your PC a claim, especially if Blades back you up. The big problem/boon is that for regular man and mer on tamriel, Septim and Dragonborn became interchangeable over the years.