When it comes to the Night King and Voldemort there is no way to definitively say that they are the same character. After all they exist in different universes and very different stories. But if you go beyond the story line and look at what they actually are, you might find a much stronger connection. They both do share a similar trait since they are both magic users, they are both immensely powerful, and they are both, technically, liches that have given up their humanity for something that wholly evil and impure.
Let’s pause for a moment and reflect on the fact that the Night King did not willingly give up his humanity as his condition was foisted upon him by the Children of the Forest. He was turned into the thing he would become, while Voldemort willingly gave up his humanity for a chance at immortality. This keeps them apart but it still unites them as the same type of character since they are both undead in a way. Such a trait is an important characteristic of a lich, as such a character is known for being something that is beyond the living.
Both of them were changed by magical means, meaning that either they were both great magic users, which is true in Voldemort’s case, or were altered somehow by powerful magic that granted them the power they now have. The power of a lich is typically attached to something or someone that acts as a phylactery, a holding place for the lich’s soul. For Voldemort it is the horcruxes, but we have yet to see if the Night King has any such weakness. There is a thought that perhaps the grove where he was changed could hold some secret that might explain his continued existence or hold great importance for him, but thus far the idea that he is in any way vulnerable in this manner has not been fully explored.
Another trademark of the lich is that they are pure evil. Even if they were a decent person in life the act of being brought back in this manner would create something that was completely unholy and bent only on the eradication of life. Voldemort was the epitome of burgeoning evil when he was younger and didn’t get any better as the years went by. It isn’t fully known yet if the Night King was a good man or not but what he came back as was something entirely devoted to destruction and death. Thus far those that have stood in his path have felt the wintry touch of his forces and perished since they chose to fight. But if if he is coming for all Westeros or just the Riverlands where he will no doubt seek out the Children of the Forest yet is unknown.
What we do know is that in terms of what they are, not who they are, Voldemort and the Night King are very much the same type of character with only a few noticeable differences.
