That's a really serious matter. As hosts, we need to do what is best for ourselves and our listings. In my opinion, the #1 solution would definitely be to get Airbnb involved. So don't stop sending emails, remind them about your case once or twice a day if you have to.
If all else fails, and you really don't want to cancel this guest's booking, then I recommend that you meet this person for check-in. If this person misidentified themselves, then I think that would be the only reasonable time to cancel the booking. Make sure you document the whole thing so that you can inform Airbnb about this.
Hope this piece of advise helps.