I can see where an animal can provide emotional support. I think pets are good for us. Take your cat to Petsmart? Normal. Have it in your apartment? It is not bothering anyone. Moving to another state and need to take the pet on the plane with you? So long as the pet behaves, that is to be expected.
It's the people who want to take their non-service animals to the mall, restaurants, the movies, ice skating rink, the museum, etc. that irritate me. People who are allergic to these animals have just as much right to NOT have the animal around, not to mention those who have overwhelming fear of animals. They have emotional needs too. As I said, it is a balancing act. I love my pets and have had pets my entire life. But I believe non-service animals belong at home for the most part. In our culture, we do not let cats and dogs go everywhere and anywhere.
Under my belief in balancing, Person A wants to take their pet someplace where pets are not regularly seen so they can have the emotional support of the animal. Person B is phobic (or severely allergic) about the animal in question and expects to be able to visit the place without having to be in close contact with such an animal because animals really aren't supposed to be there. Who trumps? I say the ones who expect to be in a place where animals really do not belong. (Service animals excepted.)
Truly, you hardly ever see this with cats. It's usually someone dragging their pocket dog EVERYWHERE. I miss the days when dogs were dogs, people were people, and humans had enough sense to know the difference. Your dog does NOT feel compelled to rush to Macy's in time to buy a bracelet on sale. It does not feel incomplete if it fails to make a
Walmart run. It will not be depressed if it misses the Monet exhibit at the museum. It's a dog. And you can be separated from it for brief periods of time. It's not that I dislike pocket dogs. My mother always had one. But the dog stayed home when we went out. And it wagged its little tail when we returned. Like a dog.