It's a government requirement in India that hotels store details of foreign visitors' passports and visas, and in practice that means they always take a photocopy or scan the documents into the hotel software (Opera has a built-in function for this, I think). (Years ago they would instead laboriously copy the details into a voluminous ledger.) You can generally not bypass this by attempting to show a different id if you look visibly non-Indian, although of course at low-end places they may just turn a blind eye to all this and avoid the paperwork entirely.

In Europe, as other posters above have reported, things vary by country, often they ask to see an id, and rarely make a scan or photocopy. On the whole I've grown inured to all of it, deeming the overall risk of misuse from a hotel's copy of my id to be fairly low and the bother of trying to counter it too high.

I'm genuinely curious if people have experienced unfortunate incidents of identity theft or other misuse of their id directly traceable to a hotel's possession of copies of the id? Never heard of it in years of travel amongst my colleagues. (Doesn't mean I don't worry about it as a theoretical possibility.)