It varies in the US. Some is probably locality. Some is probably chain policy. Although hotels don't in general do the copy of passport things that is common in many places.

A few years back, I got in a tight situation. I had a last minute trip and, among other things, didn't grab the little travel folder which I have with backup ID, credit cards, etc.

Get my license out in the back of the car for TSA. Somehow I managed to lose it between there and the airport doors. Called the driver. They looked no luck. No idea what happened.

To my considerable surprise, the TSA let me fly with no real hassle. (Rules may have tightened.)

But when I got to the crappy Travelodge motel on the other side--no luck. I assume I would have had better luck at one of my regular chains but they didn't want to budge in spite of a wallet full of other cards and a photo company ID.

I wasn't sure where I was going to sleep but they eventually relented so long as I paid cash. Fortunately the cost was relatively low enough I was able to get enough of an ATM. (I didn't have photos on my phone.)