Player "owns" the PC, it's their character after all. They spent the time building it, playing it, advancing it as they decided; they shaped it into what they wanted them to be. DM has no right to it, unless the player gives them permission to use them somewhere else as an NPC in a quiet town or something, possibly even a tether to another campaign. I say this because growing up, I watched my dad and uncles play some of the same characters they are playing in the game I'm DMing for them now, if I were to have assumed that they were completely done with them and put them into the game, it would not only have been awkward, but I don't honestly know how they would have felt about that. As the DM I get control, or "ownership", over the world, the plot, the history of it, the story, the people in it, and I get to shape that world however I see fit; players only get to control a small drop of that through their character and their backstory, that's if you want to go based purely on who created what. I will be honest, this is an odd question to be asking, as the only reason I see the need to establish ownership is if someone is trying to write a book or something with someone else's creative IP. Outside of that, it seems like a no-brainer to say that the player owns their character, even after the game is done.
EDIT: Some spelling errors and a mix up on player instead of PC