@Elena3677 Unfortunately Airbnb only requires an email and phone number. I have come across quite a few profiles that have very obvious fake names. Ex a recent guest identified as DJ Echoz. Obviously that was a stage name, but it goes to show Airbnb didn't even have a requirement for his real name to be submitted.
That being said we as host have to do our due diligence in filtering potential bad apples. Past reviews provide some indication of a guests behaviors when renting. Personally I require guest to submit a government ID to be verified through Airbnb. Although an ID does not define their character, it gives me some comfort knowing they are who they list themselves to be, and not trying to present under an alias.
As for background checks, its at the guests' discretion to consent to providing the necessary information for it to be completed. I can see both sides. as a host we want to feel safe allowing a stranger into our property, but as a guest it might feel like an invasion of their privacy to submit to a more invasive screening for the purpose of a short term rental.
In the end , I would at a minimum expect the guest to provide a full name, for themselves, and those in their party. With a full name you could likely do a basic search. Depending on your location, a lot of records are public. If she is declining to do that, I would see that as an issue. It might be best that she cancelled.