@Marc715
you can do it but you have to have it in writing in your house rules.
In Croatia, Italy, Spain, and some other countries are mandatory to take personal information from all our guests, including infants, and to upload that information to police immigration department's online application/form. I would do it even if it wouldn't be mandatory for the same reason you experienced today.
I can't believe how many hosts give the keys of their properties to total strangers they don't know anything about. In Airbnb, TOS is written that Airbnb will not reveal the guest's information to anyone. The only thing host can do in such a situation is to call the police and then let police or the court to ask Airbnb for guest's ID. Immagine how much time is needed for this process!
Some hosts say in certain countries hosts are not allowed to identify their guests but I can't believe it is true because, as you said, every hotel and hostel and every landlord requires identification from their guests and tenants so why shouldn't you?