@Marcella185 Airbnb is constantly changing its policy on assistance animals (literally, several times per year) so you have to frequently re-check this link to see where you stand: https://www.airbnb.com/help/article/1869/accessibility-policy
The language in the policy used to say that hosts could be exempt if they shared living space with the guests, but now they've switched to a more vague statement:
"A Host may qualify for an exemption in certain circumstances — for instance, if the service animal directly threatens their health or safety."
Whether they consider an allergy to residual dander to be a health and safety threat is unclear. But actually, what you should be really concerned about is this part:
"Guests are allowed to be accompanied by service animals during a stay or Experience and are not required to disclose the presence of a service animal before booking. "
So anyone who books your home on Airbnb can just show up with their pets without warning, call them "service animals," and there's nothing you can do about it. If there's any chance that this could be a serious health risk to you, RUN, DON'T WALK away from AIrbnb. They're not going to spot your medical fees if an allergy attack lands you in the hospital because of their stupid policy.