I’m not going to respond with a lot of depth because I don’t think I have a good enough understanding here to be particularly helpful, but I suspect a collar would be considered as inappropriate in a school because of its strong association with BDSM practices alongside the general societal expectation that one’s sexuality is kept out of the public eye. I think that notion can also apply more broadly to the situation as a whole, at least to an outside observer.
You’re generally required to provide identification if you’re arrested and in some US states police can compel you to identify yourself under certain other circumstances, but otherwise yes, in the US you are never required to talk to police beyond this.