I think the comparison to the propiska system is incorrect. This Soviet system heavily controlled internal migration and was what ultimately dictated where someone was permitted to live. You couldn't relocate without one, and having this permission was tied to all sorts of local services. This system anchored people to where they were, and usually barred them from moving unless they had a good reason to.

The US currently has freedom of movement. You don't need the government's permission to live somewhere or to move somewhere else. An ID with your address listed isn't propiska. At best, you could compare it to the 'internal passport' that the USSR and most post-Soviet countries had, which acted as a comprehensive identity document and was the ancestor to modern national ID cards that are used in many countries.