The sale contract with the dealership probably has some language like "contingent on final approval from the loan company," and the lender probably has their own rules about only approving loans for people with a valid driver's license. The dealership should know those rules, but I think this person is getting caught in the very unfortunate but definitely real bureaucratic nightmare where it's often your own responsibility to know that someone who you believe knows the rules better than you is wrong.
That said, everything up to the last few sentences sounds like they're trying to do things the right way, but if they try using One Weird Trick(tm) to keep possession of the car they're probably going to cause much bigger problems for their future self.