When cosigning for a car and the person who is acquiring the car is the only one making payments whose name is the ownership of the car under?

the person making the payments is actually buying the car. it is illogical that a cosigner could claim any ownership rights, having paid nothing. the person having made the payments is the owner It depends on the title. If it is only your name then it is your's. If it is both names, then it belongs to both of you regardless of who pays.