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I could find two primary sources about the Dragon Born. http://www.imperial-library.info/content/songs-skyrim which describes the Dragon Born as a hero from prophesy. http://www.imperial-library.info/content/book-dragonborn written in 3E360 which describes the Prophecy of the Dragonborn and how the word relates to the Septim Emperors and their predecessors. If there is anything I'm missing, feel free to add and comment.
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I'll be making some arguments that assume you've at least read these two books, the PGE, brief history to the Empire, ect. But feel free to ask for a reference.
Dragonborn Emperors
Those who become Emperor and light the Dragonfires are surely Dragonborn - the proof is in the wearing of the Amulet and the lighting of the Fires. But were they Dragonborn and thus able to do these things - or was the doing the sign of the blessing of Akatosh descending upon them? All that we can say is that it is both, and neither - a divine mystery. - http://www.imperial-library.info/content/book-dragonborn
The book claims that being dragonborn is not hereditary and means being blessed by Akatosh. Though it is not clear about it whether this is a pre-existing condition or something that can be acquired by going through the ritual of lighting the dragon fires. When I read this though, I'm thinking that the priest is trying to provide divine legitimacy to the rather mundane rules of the royal succession and can't deal with the resulting circular logic.
Reman (The Cyrodiil): Culture god-hero of the Second Empire, Reman was the greatest hero of the Akaviri Trouble. Indeed, he convinced the invaders to help him build his own empire, and conquered all of Tamriel except for Morrowind. He instituted the rites of becoming Emperor, which included the ritual geas to the Amulet of Kings, a soulgem of immense power. His Dynasty was ended by the Dunmeri Morag Tong at the end of the first era. Also called the Worldly God. - http://www.imperial-library.info/content/varieties-faith-empire.
Looking at some older material that predates the concept of the Dragonborn and Dragonfires, there was a ritual geas to the Amulet of Kings. So from that I reckon that going through the ritual and creating the Convent, repeating what Alessia did, should be enough.
Who gets to be Dragonborn?
So long as the dragonfires shall burn, to you, and to all generations, I swear that my Heart's blood shall hold fast the Gates of Oblivion.
So long as the Blood of the Dragon runs strong in her rulers, the glory of the Empire shall extend in unbroken years. But should the dragonfires fail, and should no heir of our joined blood wear the Amulet of Kings, then shall the Empire descend into darkness, and the Demon Lords of Misrule shall govern the land.' - http://www.imperial-library.info/content/trials-saint-alessia
So long as the Empire shall maintain its worship of Akatosh and his kin, and so long as Alessia's heirs shall bear the Amulet of Kings, Akatosh and his divine kin maintain a strong barrier between Tamriel and Oblivion, so that mortal man need never again fear the devastating summoned hosts of the Daedra Lords. - http://www.imperial-library.info/content/amulet-kings-0
This relates to a much older problem. In Oblivion the player could not Wear the Amulet of Kings. And through the Trials of Saint Alessia and the Amulet of Kings, the story was told that only true heirs could wear the Amulet of Kings.
Underneath this was a little mystery. Mankar Camoran was wearing the Amulet of Kings and he was not a true heir. This mystery could be solved by realizing from the proper sources that the Amulet of Kings was in fact and Ayleid artifact and that Mankar Camoran came from another line of Kings and was a rightful heir to the Amulet. If anybody is interested in this in more detail, feel free to make a new thread.
However this little mystery got a little lost in the fray after Oblivion. People had spend allot of time charting out the Imperial lineage and noted that it was full of jumps and branches and this created more then a few plot holes. It became hard to believe that the Amulet of Kings had to be inherited. It also became hard to believe that every heir to the throne had been killed. If anybody is interested in this in more detail, feel free to make a new thread.
The result appears to have been that the blood relation was no longer important. The book of the Dragon Born uses the same arguments that were used in those discussions. Now having the Dragon Blood means merely being blessed by Akatosh and allows you to wear the Amulet of Kings. But this again creates a problem with Mankar Camoran. Mankar wore the Amulet of Kings, something only those with the Dragon Blood should be able to do, something only those blessed by Akatosh should be able to do. Yet it seems strange Akatosh would bless the one person who'd bring down the walls he had sworn to maintain.
A priest might say that the gods work in mysterous ways. Knowing that the Amulet of Kings is just an Ayleid artefact, it makes more sense to assume that the blessing of Akatosh is not relevant. There might be some mechanism as the player can not wear the Amulet, but it is neither divine nor hereditary.
If anybody wants to speculate on that, again something that would deserve it's own thread.
To summarize the way I see it: The Amulet of Kings is merely an Ayleid artifact. Akatosh blessing is not required to wear it. Rather the importance lies in the ritual of lighting the Dragonfires, the ritual of activating the barrier between Mundus and Oblivion. This ritual may or may not be conducted with Akatosh help and blessing.
Dragonborn Heros
There isn't much to Songs of Skyrim, but the Dragonborn there is a hero. And Book of the Dragonborn says the same: The Nords tell tales of Dragonborn heroes who were great dragonslayers, able to steal the power of the dragons they killed. As simple as this is, it is also almost completely at odds with what we know about the Dragonborn Kings. Of all emperors only Tiber Septim was able to use the Thu'um.
And if anybody wants to discuss the Heresy, now would be a good time to make thread for that. Who did the actuall shouting won't be relevant for this discussion.
Most scholars agree that the term was first used in connection with the Covenant of Akatosh, when the blessed St. Alessia was given the Amulet of Kings and the Dragonfires in the Temple of the One were first lit. "Akatosh, looking with pity upon the plight of men, drew precious blood from his own heart, and blessed St. Alessia with this blood of Dragons, and made a Covenant that so long as Alessia's generations were true to the dragon blood, Akatosh would endeavor to seal tight the Gates of Oblivion, and to deny the armies of daedra and undead to their enemies, the Daedra-loving Ayleids." Those blessed by Akatosh with "the dragon blood" became known more simply as Dragonborn. - http://www.imperial-library.info/content/book-dragonborn
Odly enough, the term does not appear in either Trials of St. Alessia or the Amulet of Kings. The only other description of Alessia comes from the http://www.imperial-library.info/content/remanada-chapter-1-sancre-tor-and-birth-reman. "And to this host appeared at last a spirit who resembled none other than El-Estia, queen of ancienttimes, who bore in her left hand the dragonfire of the aka-tosh and in her right hand the jewels of the covenant and on her briast a wound that spilt void onto her mangled feet." The Dragonfires, the burning blood of Akatosh, is the relevant imagery.
Now the argument can be made that the term was not used because the material I've been referencing is from Oblivion rather then Skyrim you might argue retcon. I'm going to do something much more interesting.
The only overlap between the Dragonborn heros and Dragonborn kings is Tiber Septim. Tiber Septim was also the Emperor to publish the First Pocket Guide to the Empire. The PGE is a propaganda piece. On one hand it vilifies the Altmeri Dominion with which Tiber will soon go to war. On the other hand it glorifies the Nords of Skyrim as slayers of Mer and suggest that Tiber was one of their legends, master of the Thu'um, born in Altmora. He is a true nord, dragonborn and he is preparing his call to war.
I don't know if it is true. When you complete your training with the Greybeard's they say something along the lines of "You have become now Ysmir, Dragon of the North". If this true or not is a discussion for another thread, what matters here is that this is the image Tiber Septim creates of himself.
Divine Rights of Kings
There is a medieval theory called divine rights of kings. It poses that god gave worldly powers to the kings and eclastical powers to the church and if he wanted it to be any other way it would not be so. This theory was used by Kings to legitimise their rule. It became their divine right to rule.
What Tiber did with the first PGE was equivalent. He legitimised his rule over Skyrim. But this poses a bit of a problem for the dynasty. They do not have the Thu'um. They are not Dragonborn. They are not from ancient Altmora. They don't even descend from Tiber Septim!
So I believe that concept of the Dragonborn was married to the Dragonfires and the Amulet of Kings to legitimize the rule of Cyrodiils Emperors over Skyrim. Thematically not too difficult as one can be in possession of Dragon blood by having it in their veins or in their hand.
What I have not yet covered and can't realy find much material for is why appearing to be Dragonborn, why appearing to be a dragonslaying hero with dragon blood, is something you can use as a basis for the right to rule. For this I'd use http://www.imperial-library.info/content/five-hundred-mighty-companions-or-thereabouts-ysgramor-returned to suggest that Nords were Dragons that and being Dragonborn makes you a true Nord. Or perhaps argue that having the Dragonsblood makes you a bit of a Dragon, once the legitimate rulers of Nords.
Summary
tl;dr?
The Amulet of Kings is an Ayleid artifact and Akatosh blessing is not required to wear it. Rather the importance lies in the ritual of lighting the Dragonfires, the ritual of activating the barrier between Mundus and Oblivion. This ritual is associated with the the Amulet of Kings and the Dragonfires, symbolizing Akatosh burning blood. To legitimize Cyrodiils rule over Skyrim after Tiber Septim, the concept of the Dragonborn, being in possesion of Dragonblood, was married to the Dragonfires, being in possesion of Akatoshs blood.