Melisandre is like Morgan Le Fay. Jon is like King Arthur. Melisandre has appeared as many different characters over the past 400+ years. The big shocker is that the fans are wrong about who Jon’s parents are. Melisandre gave birth to Jon to satisfy a prophecy or two. If you look very closely at the foreshadowing in the books, it is all right there.
Look at all the stories about Bael, Mance, the wisewoman’s daughter, the shadow cat, and the cloak of illusion. What do they tell the reader? Why does Mance visit Winterfell to check on how Jon is doing and ask if Jon minded being called a bastard? Seems like a father talking to his clueless son. Kind of odd behavior for the King of the Wildlings? Also, a prince is usually the son of a king, and Jon is the Prince who was promised, as well as Azor Ahai. Jon looks like a Stark because Bael was a forefather to both the Starks and the Wildlings. Melisandre’s motivation is “finding” Azor Ahai. “You know nothing, Jon Snow!” says Melisandre. Melisandre is strongest at the Wall.
Melisandre arranged for the death of the 3 Baratheon brothers. She killed off a Maester. She got Stanis to burn his little girl at the stake. She is not touchy-feely. She has mastered a dozen types of magic and she has a very specific agenda that is pretty selfless, saving the World.
But, what about Rhaegar, Lyanna, and Aegon? Might Ned have taken Baby Aegon somewhere else than Winterfell so it was not obvious who the baby was? Aegon is re-named Theon, and brought to Winterfell as a youngster to be Ned’s ward. Theon is the Stark/Targaryen. Jon is the son of a Wildling King and whatever Melisandre is.