It's called an ID holder. It is designed to hold the ID. The correct procedure when presenting an identification card is to physically hand the card over to the person who needs to inspect it. This stems from the tradition of a visiting emissary presenting his credentials to a king as proof of identity. The document was physically handed over to the person who verified its authenticity.

As pointed out by thegeneral, try getting cute with a police officer bickering over an ID holder and see what happens.