I realise as a continental European you are brought up to accept the notion that the cops can stop you and demand ID at any time, but to those of us in the Anglo-Saxon world that just aint the case and most of us will fight for it never to be the case (despite what the crooks in Brussels try and make us do). If you took a moment to think about your statement on carrying "proof of status" in the US, I think you'd realise it is nonsense as a bone-fide US citizen or resident by definition wouldn't have to have that ID, so how could the cops tell them apart from the "aliens"?
Before taking pot-shots at the US, I suggest you look a little closer to home. The whole concept of Anmeldung is something a totalitarian state would be proud of. And as for the Auslaender Gesetz......