I suppose it's possible that he could have shown them a message from someone who has the same first name - after all how many people do you have in your phone as their full contact details? Or he could have just changed the contact name of someone else to her name, so all the messages who were originally from 'Amy' would have looked like they were from 'Sarah' or whatever. Or he could have a spare phone/work phone, added that as a contact and sent a fake message saying 'come up to my room.'
Or they could be lying out their arses and there were no text messages.
TBH I think it's a red herring. If one person books a room nobody should be given a key, no matter how well they know the person. A woman escaping DV might go to a hotel, obviously her partner would have more than a few text messages from her and would know all her other details - he still shouldn't be allowed in without checking with her that she wants him there! It's so stupid because if someone has their phone and the person's phone number then surely it's much easier to say 'Well give them a ring and ask them to let you in, then!'
There are so many security risks - as pp have said he could have murdered her, but (although not as serious as what did happen) - if they're letting people have keys solely on the basis of knowing their name anyone could access the room while someone is out and steal all their stuff as well. So not even a woman/lone traveller risk but a risk to anybody.