People in Soviet Union (and in Russia, still) have to carry an internal passport (distinct from the regular passport, that is called smth like "foreign-travel passport" in Russian). The cops can legally stop you on the street and ask for ID. If you don't present your internal passport (or similar), they may take you to the police station "to determine your identity" at their discretion. Happened to me personally with an expired internal passport on my literal birthday that it expired on, I was about to go to a police station cause I didn't have much cash on me for a bribe, but they decided to let me go because hey, it's my birthday.

Vaccine cards (I am vaccinated and boosted and would have done it based on my own decision making) feel significantly similar to me. As in Russia, when dealing with totalitarian aspects of government, I find it easier to just comply, but if I had more free time on my hands I would refuse to show the card on principle. Needless to say politicians advocating for these would never get my vote...