I agree the owner is being registered rather than the aircraft. We are not "pilots" (except for those who are) so what do we call ourselves? But I can allow my unregistered son to fly my quad. This is different from a car in traffic, where my son who does not have a license cannot drive on public roads, period, even if I own the car and am in it with him. So it's not exactly like any other licensing situation I can think of.
And if you don't have your registration handy when a cop pulls over your car, the cop can just call his dispatcher and check that information and more. Hopefully they could do the same in case we're operating without a copy of our certificate.
Actually, the car being registered and your son not having a license is completely unrelated. He could take that car out for a spin and face the consequences if he's caught. The consequence YOU would face would be as a parent, not a car owner. It's actually exactly the same in this manner.
If he crashed, you would then have to face the consequences as the car owner and it's the same with him flying the quad. As I mentioned previously, the wisdom of allowing someone else to take a spin without a license is questionable, but the state of registration of the vehicle and of it's operator are entirely different. It's not about whether you WOULD let your son drive your car without a license, it's that he COULD, if he were so inclined.
Each aircraft we put our registration number on becomes a registered aircraft that's now tied to us. As it is for your car to be legally on the road (not the driver, mind you), for that quad to fly around registration needs to be in order.