@Rosemary12, I have had a few guests now with just the initial or a middle name for a last name. They are always young women, and it is always because of the Facebook sign-up. They are worried about their own security on that platform, which is understandable. In every case it has been fine, but I get why you are worried. You are sharing your home. Your security is important.
Don't feel badly about asking a guest for her full name, or for the names of the people in her party. It is well within your right to ask, and she may not have realized her full name has not been given to you automatically. Guests are a bit in the dark about those things - some think I have seen their drivers' licences!
When the system won't do what you want, you are doing the right thing by taking it into your own hands. "Hello, Miss M. May I have your last name? Airbnb is built upon trust, and I don't know who you are!"