If it’s a shared space listing (they’re renting a room in your house) you don’t have to take either the fake “emotional support” animals or service animals. The Fair Housing Act in the US exempts owner occupied shared housing from the service dog rule, and Air’s rules let you not allow pets or kids in a shared listing.
If it’s not a shared listing, you CAN require that the animal not be left alone, and that it always stay with the person it’s “supporting”. Some hosts require that they be kenneled.
Air’s stupid animal rules are one reason why I would never have a full house rental.