People are using all kinds of animals as service animals. One fellow came home from the Middle East and now uses a goose as his service animal, but he knows everyone's limitations of acceptance and expectations and abides by them. When I worked for United, we had almost all dogs and a few cats. Flying in and out of LA and New York, we flew a lot of personalities and the women, (mostly, but not always), had to have their exotic dog. There were also a lot of people that just brought Fluffy and Mittens on-board and paid the premium to stuff them under a seat. I flew Anna Nichole Smith once from LA to New York to appear on Letterman and she had her exotic pet dog with her. (Sorry, I do not know my breeds, but it was one of those really little dogs that yap all the time.) Her and the dog each had their own seats in First Class. Can you imagine paying around $2500.00 to fly your pet dog? It's a five plus hour flight, so the dog will have to go to the bathroom. Her dog was diapered and fed on-board much to the displeasure of the other passengers sitting in First Class.
I have several pet and service animal stories. Over my 34 year career, I have seen a lot, but I am sure that I haven't seen it all. My First Officer and I were sitting in the flight crew's lounge in McCarran Airport in Las Vegas, just having a cup of java and relaxing before our flight to LAX. My F/O was looking over the passenger manifest and starts laughing. I asked him what's so funny. He said that Jack Benny and his wife are flying with us and his wife is also bringing their two Shitzu's. (Spelling?) I wasn't sure what to expect, but hey, it was only a one hour flight, so I wasn't expecting any issues and there weren't. This was back in the day when we still had the door to the flight deck open at times. I was able to see back into First Class where the Benny's were sitting and they were surrounded by people, laughing and just having a really good time.
OTOH, we were flying from Albuquerque to San Francisco and we had a man and lady traveling with their Pug. This dog must have disliked everyone except his family. He barked almost the entire trip at any noise or person that walked by him. Not a good trip at all. We used to talk about this all the time. What do we do when passengers complain about service animals? It's a Catch 22 situation, or sort of, damned if you do and damned if you don't. Our poor F/A's used to hear, "If you don't do something about that animal (pick one), I am never going to fly United again." We try to tell them that we have to comply with the law. If we knew ahead of time that the animal wasn't going to behave, we could forbid it's entry into the plane, but we don't know that until we get airborne.