In the US, I'm not sure I've ever had a desk worker more than glance at an ID. Many places do require one though--had a heck of a time checking into a Travelodge of all things once because I had lost my license. Not sure how much of this is hotel rules and how much local regulation.

Makes me wonder how this works when you use digital check-in.

It's true in Europe they often copy your license and, in some countries, I believe there's some registration with the police that the hotel handles.