Honestly, i've heard a lot of horror storries on Airbnb about hosts (not in terms of ID theft, but just false advertising, unreliability, etc), so i'd be careful, because their verification is not that thorough at all.
I got lucky that i rented directly from a property management company, so i didn't feel like it was an issue to give them a copy of my ID. I also had to give a copy of my ID to my first landlord for my first apartment, which i booked on agoda. I was initially hesitant, but they had many reviews from guests so it seemed safe to assume they were legit.
It's pretty standard to provide a copy of your ID to a landlord, but i would always meet them first and try to read whether they seemed trustworthy. I would not give them a copy of my COPR, you don't have an obligation to provide evidence of your immigration status from what i understand. At most if they wanted to see it i could show them, but i would not allow them to make a copy.
I also don't usually give a copy of my passport, i have a national ID from my home county, i prefer to give that because good luck to them trying to do anything with it lol. I know not everyone has one in their country however, so they have to provide their passport.