Lets say the police swing by your hunting campsite, they are friendly, you are friendly, they check your license, guns, ammo, whatever...
Its all good, everything is in order, right before they starts to leave, one of the cops turns and says "WAIT! Where is that piece of paper that says "keep this paper!" DUN DUN DUNNNNNNNNNNN! Better call Sol!
Look, here is the bottom line, if the cop is so miserable, angry, bad, as to hassle you over this piece of paper that literally provides ZERO information then guess what? He was going to hassle you over you magazine limits, barrel lengths, storage cases, and every other little thing. Your day was already going to be spoiled with or without that pointless paper. We all know the firearms laws are written purposefully so there is basically always something they can hassle you for. If the cop is a reasonable nice guy, so are you (that is very important), and noting else was obviously wrong, then he isn't going to give a rats a$$ about a piece of paper that he probably doesn't even know about to begin with.
OP I understand you being curious about the paper... you will find there are many other laws and "things" surrounding firearms that dont make a whole lot of sense. I find following the law as best you can and being polite with "the law" is the best bet. If you lose the paper, don't worry about it. In a two or three years it will disintegrate in you wallet anyway. Horse officially beaten to death.