Airbnb can and does have the ability to set its own rules for users. If you object to their rules, you can certainly protest them or take them to small claims court to challenge them in a court of law, but basically, if you don’t want to abide by the rules of any business, you are free not to use their services.
For instance, there is nothing illegal about telling friends and family that they can’t bring their dog, even a service dog, to your personal home, because you may have allergies or be afraid of dogs, or it traumatizes your cats, but Airbnb can and will ban users who refuse to host a guest because they have a service dog.
There are no ADA-type laws in Mexico, where I am located, saying you can’t refuse to host someone with a service animal, but that doesn’t mean Airbnb couldn’t choose to delist me if I refused a guest on that basis, because it violates their policies.
As for whether a space is considered to be private or public, I’m not sure whether there have ever been any legal cases that conclusively defines those terms.
Certainly a public space, like a library or a town square or park is public in a way that a private residence is not. Just because a host may have a hallway that multiple guests and the host uses doesn’t necessarily mean it would be legally classified as “public”. The very fact that only the host and the guests have access to the home via a key or coded entry device, rather than anyone being able to walk in off the street, seems to me that whether it is a public or private space is a grey area, unless there are legal definitions of what makes an area public or private.
What would be interesting would be a court challenge regarding Airbnb considering a space private in some instances, yet public in others. I don’t see how they could win a legal argument that they consider a hallway in a host’s home to be a private space where recording devices aren’t allowed, yet consider the same hallway to be a public space where service animals must be allowed according to ADA regulations.