As already mentioned, the official and canon answer to this is, "We do not know yet". However, it can be speculated that the NK is not a Targaryen.
Flawed Premsie: Targaryen identity
Controlling a dragon doesn't make you a Targaryen. There are non-Targaryen dragonriders historically such as:
- Emperor Aurion
- Jaenara Belaerys
- Addam, Jacaerys, Lucerys, Joffrey1, Laena, Laenor2 Velaryons.
You don't need Targaryen blood to ride a dragon. It doesn't help much anyways.
The prophecy speaks of three heads of the dragon. It doesn't mention anything that the dragons have to be patrilineally descended from House Targaryen or have to be dragonriders.
The woman asks him if he will make a song for the child, and he
replies that he has a song and that "He is the prince that was
promised, and his is the song of ice and fire.". He appears to look at
Dany then, as if seeing her, and then he adds that "There must be one
more," and "The dragon has three heads.".
However, since the Targaryens were last prominent Dragonlords (Others being the Velaryons and the Celtigars), things got kind of mixed up and more centered on legacy of Targaryens.
Also note that Rhaegar who said that is known to have mistaken before about the meaning of the prophecy twice. At first he thought it was himself. Then he thought it must have been his son Aegon. He was wrong.
Night's King actually predates the foundation of Valyria, as per the books. So he can't possibly be a lost Targaryen. He existed during the Age of Heroes (ca.-8,000 or -6,000 BC) while Valyrians established themselves 5000 years before Aegon's conquest. I don't know why you'd think he was a Stark either, his true name is lost. Sure there is implication that he may have been a Stark but that's not confirmed.
- Books' NK vs Show's NK
- Difference between Others and Wights - Note how Others can reanimate the dead and control them. Can they do the same with living dragons?
- Can Wargs control dragons? - Since we are on subject of controlling dragons by means other than bloodmagic.
- Westerosi Dragons and their riders - Notice the non-Targaryen riders
1. Jacaerys, Lucerys and Joffrey were half Targaryens; sons of Princess Rhaenyra Targaryen
2. Laena and Laenor were half Targaryens; children of Princess Rhaenys Targaryen