The U.S. Is complicated by the fact that each state decides how to handle license plates. Sometimes, the plates stay with the car. Sometimes with the owner.
In New York, the plates effectively remain the property of the DMV.
In New York, the plate is issued with the car registration. You keep that plate as long as you own the vehicle, and the registration remains valid, unless the state decides to issue a new design of plates. When that happens, you may be allowed to leave the old plate on the vehicle, or, depending on the political mood at the time, and the state's budgetary needs, you may be forced to change to the new design. Right now, there are 3 basic series of plates that are legal (not counting styles of vanity plates.) If you change then you will be instructed to destroy the old plates. If you do not, and someone uses them illegally, you could be held responsible.
If you sell or otherwise change ownership of the vehicle without purchasing a replacement at the same time, you must surrender the plates to the DMV in order to cancel your insurance and registration on the vehicle.
If you have a replacement vehicle, you may transfer the plates to it, providing the DMV with all the proper paperwork.
There's probably another 40 paragraphs in the law, but these are the basics.
And this is just one state of 50.