I’ve been quite successful in asking for this information, including a screen shot of a Driver’s License or passport. I say - truthfully - that I need it for insurance purposes. So far, so good and no issues with Air.

Are the rules the same with Airbnb as they were with VRBO as stated above?
Message the guests through the Air platform. Tell them they have violated not only your rules, but also the rules of the HOA and your insurance. Then immediately call Air and cancel their stay, referencing your message to them about the HOA rules.

f the guest refused to provide me with names then I would ask Airbnb to cancel as “uncomfortable with reservation.” As an IB host you get penalty free cancellations. I don’t know if the HOA rules would give you leverage with Air or not.
I have done this with IB guests who have 0 reviews and have not answered specific questions and then tried to book with children or pets. Air has canceled penalty-free.