Who is Azor Azhai? Was Melisandre's entire life a lie? Throughout the show she has put forward:
But
Arya Stark ends up to be the one to end the Night King. Clearly she thinks that her purpose is over because she died after the battle, so the Night King isn't coming back anytime soon.
Who is the Prince who was Promised in light of the
NK's death?
Assuming there has to be a Prince that was Promised, which there doesn't, especially in the show, then the following is what is probably intended.
It is certainly not Stannis, Melisandre admits she was wrong there.
DAVOS: The only way for what? They all died anyway! You told everyone Stannis was the one. You had him believing it, all of them fooled. And you lied.
MELISANDRE: I didn’t lie. I was wrong.
Game of Thrones, Season 6 Episode 10, "The Winds of Winter"
She then believes it is Jon Snow
DAVOS: You serve Jon Snow now?
MELISANDRE: He’s the prince that was promised.
Game of Thrones, Season 6 Episode 4, "Book of the Stranger"
Later on she states that Jon and Dany have a part to play but doesn't state either are the prince/princess that were promised.
MELISANDRE: The Long Night is coming. Only the prince who was promised can bring the dawn.
DAENERYS: The prince who was promised will bring the dawn. I'm afraid I'm not a prince.
MISSANDEI: Your Grace, forgive me, but your translation is not quite accurate. That noun has no gender in High Valyrian, so the proper translation for that prophecy would be the prince or princess who was promised will bring the dawn.
[...]
DAENERYS: And you believe this prophecy refers to me?
MELISANDRE: Prophecies are dangerous things. I believe you have a role to play, as does another. The King in the North, Jon Snow.
Game of Thrones, Season 7 Episode 2, "Stormborn"
Note at this point that D&D would have known they wanted Arya to kill the Night King as they stated in the Inside the Episode for Season 8 Episode 3 that they've known she would be the one for about 3 years so this was foreshadowing for her not Dany.
As such I think we can confidently state that with these events and that Arya actually kills the Night King that Arya was the Prince that was Promised.
In the books, everything points at Daenerys being Azor Ahai.
The legendary Azor Ahai, a messiah figure from the "Red God" R'hllor's faith. He had to stick a sword through his beloved's heart for the sword to become magical, Lightbringer. Dany does something very similar when she burns Khal Drogo on a pyre and thereby obtains the dragons. So the dragons might be Lightbringer, since Dany "drew them from the fire":
There will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world. In this dread hour a warrior shall draw from the fire a burning sword. And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be Azor Ahai come again, and the darkness shall flee before him.
There is a red comet in the books and series both which indicates the arrival of Azor Ahai, after the birth of the dragons.
Another prophecy says that Azor Ahai shall be born from salt and smoke. Melisandre believes this refers to Dragonstone and that's why she teams up with Stannis. But he wasn't born there, while Daenerys was.
The prophecy doesn't match Stannis, nor Jon Snow. Jon Snow was supposedly born:
In Dorne, in the Tower of Joy https://gameofthrones.fandom.com/wiki/Tower_of_Joy
which doesn't match the prophecy.
In the books, there is also a Red Priestess named Quaithe who believes Dany is Azor Ahai. She speaks various prophecies and riddles, predicting events in Dany's future.
So my qualified guess is that they simply screwed up the Azor Ahai prophecy completely in the series and something entirely different will happen in the books.
Did the prophecy say that the promised prince(ss) would personally kill the Night King? No, it does not. It says they will bring an end to the darkness. This does imply ending the Night King, but it doesn't specify how. Things that could qualify:
The first one would be Arya for sure. The second one could be fulfilled by Danny, or Jon, or Bran, or Theon. The third by virtually anyone, including Cersei. The thing is, the "blatantly obvious" way of ending the darkness, the first one, requires the trap and the situation that permits the trap to ever happen. If those don't happen, nor does the stabbing. Which one is really the one that resulted in ending the darkness?
And guess what: since the victor writes the history, you can be pretty sure that whoever claims final victory will also decide who "really" ended the darkness as prophesied. It's sort of like asking who wins the war: the general or the soldier? The general forms the plan, the soldier executes it. The soldier may be the one that literally kills the enemy, but the general was necessary to create that opportunity. Or was it the general's liege that won the war? His liege is the one that put him in charge, and if he hadn't done that then perhaps none of the rest would happen. Or was it the Queen's consort, who tried to challenge the enemy commander directly and failed so spectacularly that the commander became so arrogant and consumed with the idea he was unstoppable that he allowed an opening to kill him to exist in the first place?
That's the thing with prophecies. They're easy to fulfill because they're so vague.
Finally, part of the prophecies state that the promised prince(ss) would be a "three-headed dragon". This leaves open the possibility that it is not one person, but three. The House Targaryen symbol is a three-headed dragon, so some think it means a single Targaryen, but this need not be the case (remember: prophecies are vague, not specific). Furthermore, the very tales of the original Targaryen invasion involve three dragons, each ridden by a different sibling. Only one ruled as King, but all three individuals were necessary and important.
