The rental agreement lists authorized drivers. It also serves as your registration while driving the rental car. Or so I have been told when renting cars. The driver not being listed, and no listed person being present, would seem to be a suspicious circumstance suggesting possible theft of the car.
So, if my car is in my name and all applicable registration and insurance cards, and my wife drives it, there is a reasonable suspicion she stole it? Seriously? What if she kept her maiden name for professional business reasons?