In the States it is common to show a cert card if you are diving with a dive op (established business).
There seems to be an industry wide standard in our greater area for the dive op to ask:
- for a cert card,
- when your last dive was,
- and sometimes they ask how many dives you have performed.
For shore diving or diving that is independent of a dive op, you are free to dive at will with no "papers". Getting a gas fill is usually allowed with a judgement-call from the dive op. They can figure out if you are getting a fill for diving or for paint ball. I have not been asked for a Nitrox card in years.
TBH I think it's also that diving is possibly more common in the US -- you have warm-water holiday locations where people catch the diving bug. That and the litigious culture of ambulance-chasing lawyers, one can see why there's a need for cards. For the UK, diving isn't that common mainly due to the cold and we've the NHS (National Health Service) which will fix you plus the RNLI (lifeboats) and coast guard helicopters to get you to a chamber quickly and for free. Don't even need insurance, except for going abroad.