In the books, Old Nan tells the story of the Night's King, the one I mentioned in the OP. It is supposed to have happened thousands of years ago, and he was defeated by his brother Brandon the Breaker and Joramun. The thing is, in the books, we see that people constantly try to interpret prophecies and it bites them in the ass every time. If a prophecy comes to past, some people will say it didn't, and if one didn't actually come to past they might say it actually did.
For example, Winterfell, the Wall, Storm's End, and Hightower, are all said to have been built by "Bran the Builder", in Storm's End case it even says it was a kid, which makes no sense. Well what a coincidence that those are specifically locations that are extremely likely to be soon destroyed in the events coming (and in some cases it was the case in the show as well), at a time when we know a king Bran, a kid, will be king. So chances are it is actually a story of them being rebuilt after a great war, the one coming (in the books).
So what we see is that as seers foresaw the future and told what they had seen, over time as people repeat those stories they shift into stories about the past instead of stories about the future. And in the same way some stories that are actually about the past are retold as if they were yet to come. You can't trust stories, what people say, it's a big point in the book.
With the Night's King story, which supposedly happened thousands of years earlier, it fits almost perfectly with where the show ended, as if you could just add it to the show's ending and it would fit perfectly. Which is very strange.
Also, the book series is A Song of Ice & Fire. And the word song is constantly used as an analogy of for "sword", because stories are used like a weapon. At the end Bran says he'll look for Drogon. Why? Because the Ice threat is gone, but the fire one remains, and he wants to eliminate it as well. But if the Night's King story was to happen, that would be something Bran would want to stop; Jon would be tempted to go far north to find a lover (Ygritte or Daenerys, who knows), and become a new Night's King, and Bran would seek to defeat him, probably by taking control of Drogon to do so. There is also a possibility that the story of the Night's King is wrong, as Old Nan literally says the Night's King's name was Brandon Stark and that he may have slept in Bran's very bed, and Old Nan is never wrong. In that case Jon would be going north to find "The Three Eyed Raven"'s actual source of power and kill it. The reason the story would make Brandon the Breaker the hero is because the Three Eyed Raven has seen his own end coming and he fucked around with the story.
Also, in the books, "kinslaying" is a huge deal, you are almost guaranteed to be "cursed" if you kill family. But it seems that this has been devised precisely so that when the brothers would eventually meet (the Night's King and Brandon the Breaker), one would fear killing the other because it's supposed to be oh so abhorrent to specifically kill someone from your own family. Since The Three Eyed Raven literally implants visions in people's heads to control where the future ends up, that would have been a cultural notion borne out of his desire to protect himself from his own end at the hands of his brother.
Old Nan even says to Bran when he tells her he hates her stupid stories that "I know of a story of a boy who hated stories", well of course, because he knows his own end is coming; the Night's King stories says as much.
