While she was likely a fake SD, given the way the dog is described in the profile, the Americans with Disabilities Act does give you some guidance on these conversations, for future reference.
The US Dept of Justice says that "In situations where it is not obvious that the dog is a service animal, staff may ask only two specific questions: (1) is the dog a service animal required because of a disability? and (2) what work or task has the dog been trained to perform? Staff are not allowed to request any documentation for the dog, require that the dog demonstrate its task, or inquire about the nature of the person's disability."
You are not permitted by AirBnB policies nor by the law to turn away a disabled guest with a legitimate service animal. The DoJ further expands to say one is not permitted to charge guests for cleaning the hair or dander shed by
a service animal. However, if a guest’s service animal causes damages to a guest room, one is permitted to charge the same fee for damages as charged to other guests. Some people with disabilities may use more than one service animal to perform different tasks, so it is legal for a handler or a family to have more than one SD. The DoJ also specifically addresses the question you asked - Are guests allowed to leave their service animals in their hotel room (which applies to all lodging, not just hotels) when they
leave? A: No, the dog must be under the handler’s control at all times. The ADA is very specific that an animal who is not under the control of the handler, even if the handler is disabled and the animal is a trained SD, that is grounds to have the animal removed from the premises. SDs must be under control (voice, tether, hand signal, etc) at all times and not dangerous, destructive or disruptive. If it is not under control of the handler (which it cannot be if the handler is not there), you can insist they remove the animal or otherwise deny access to the animal, requesting the person return without the animal.
Also, big red flag, anyone who says their animal is "certified" or refers to their SD as a "pet." There is no certification for SDs in the US, UK or Canada, so anyone claiming their dog is certified likely paid for their pet to get a fraud online certification. Sometimes those fraud sites scam legit teams by making a disabled handler believe they need the certification. So it is not a guarentee that it is a faker. But, the DoJ says "Covered entities may not require documentation, such as proof that the animal has been certified, trained, or licensed as a service animal, as a condition for entry. There are individuals and organizations that sell service animal certification or registration documents online. These documents do not convey any rights under the ADA and the Department of Justice does not recognize them as proof that the dog is a service animal." Most legit handlers are well aware of this and despise those fraud certification sites. So, you have to look a little harder at anyone who claims they have a "certified" or "registered" SD.
Hope this helped some