The dragonborn is a mortal born with the soul of a dragon. Daedra are immortal beings that have existed since the creation of the world. They do not reproduce, thus it would be rather hard for one of them to be born with the soul of a dragon.

Ok I'm gonna try to arrange this so it makes sense (if possible)
Well when a Dremora dies in the Mortal Relm its soul goes back to Oblivion where its given a new body right?
And Akatosh(sp) blesses people with the soul a dragon for them to be dragonborn. And let's assume this has to be done at the time of conception for the fetus to be alive. Now back in Oblivion Dagon interrupt the Dremora souls coming back home and forces them back to the mortal realm(idk how). People who are corrupted are easily possessed so that's what happens. And one of those people just happens to be pregnant with the dragonborn. Now the dremora souls want to go back to Oblivion to get a new body but they can't so it tries to take the baby but the dragonsoul is too powerful to be destroyed and the dremora are immortal so it can't be destroyed either. Now both souls won't leave so they either do a yin/yang type battle or a soul over soul thing wit the dragonsould being the core and dremora soul being the outer layer(this makes more sense in my lecture).
Now the dragonsoul doesn affect appearances so it does what it normally does and the dremora soul changes the appearance to its liking.

AND THERE YOU GO DREMORA DRAGONBORN!!!!!!

they should just make a character with different presets each being a servant to one Daedra Prince, and you would start the game at their shrine??

First off, the Dremora aren't mindless. They are violent bastards, but they are quite sentient, and in a way noble. They also haven't always served Dagon, though their previous master is unknown.

There's actually a book out there that was written by a dremora, and it explains how the daedra in general see humans and why they behave the way they do. Basically, they are immortal, but they fear being destroyed because the period between "resurrections" is pure torment. The daedra of all shapes and sizes serve powerful masters because they believe that their masters will protect them from destruction and the Darkness.

The Daedra are antithesis to the Aedra. The Daedra are pure chaos, and the Aedra are pure order. Akatosh could no more bestow a blessing on a dremora's animus than, say, you could light water on fire with a simple match.

The Daedra are also neither good nor evil. They are above those concepts, and most of the times they simply do not care. Some of the Daedra Princes act less malevolently than others, but only because their spheres do not involve outright violence. The "good" Azura made one of her followers a lifelong shut-in just because she made a bet with Sheogorath. Azura also cursed an entire race (the Dunmer) over the actions of the Tribunal, though that is arguable. Boethiah, whom the Dunmer consider to be their ancestor and patron, is a violent bastard who makes his/her followers kill each other. Mephala, the thrid "good Daedra", is a schemer, poisoner, betrayer and in general the Elder Scrolls' answer to Tzeentch.

As Vivek said, there is really no compelling reason to worship any Aedra or Daedra.

Azura is a total [censored]. Not good at all.

As a pious Aedra worshiper and aspiring Vigilant of Stendarr, I would behead your unholy Dremora on sight.

http://youtu.be/gvdf5n-zI14

Aedra... are sort of like Daedra.
And you worship them?

Get him!