There's different kinds of service animals.

I think the ones for individuals disability can only be dogs. They must be trained in at least one particular act that can assit the person (i.e. not just bring comfort to calm anxiety). These animals can enter any and all buildings. You should register them so if asked to prove they're a service animal, you have documentation.

Then there's therapy animals. They can be cats, dogs, even horses. People bring them to hospitals or nursing facilities to help make people feel better. Unlike sevice dogs, you cannot bring them into every building. If a building doesn't allow animals, then you can't bring the animal in.

Then there's companions. All you need to have a pet as a companion is a doctor's note. Again, you can't take companions into places where animals aren't allowed. But!...No home owner can reject you because of the pet, nor can they charge you pet ddeposit or rent.

Types of working dogs | Service Dog Central

https://adata.org/publication/service-animals-booklet