This is a huge, important distinction that people fail (sometimes willingly) to recognize when they want to see an animal ban. They equate service dog with “dog”, and so laugh at the idea of a “service horse”. But…
The reason those two species are recognized is that they are the only two domesticated species that can handle the rigors of training and also the other big requirement to be a service animal – provide a service. Most people are familiar with the concept of guide dogs. They can wrap their head around an assistance dog for deaf people. But dogs also help people in wheelchairs or with mobility problems, can detect seizures or diabetic shock before the happen and alert their owner. Those are vital medical services. We don’t see the dog “working”, because (thankfully) they’re not alerting their human charge all the damn time.
But a miniature horse can accept more weight than a dog. That person may have stability issues that are better addressed by a horse than a dog. Horses, too, can perform seeing eye and retrieval tasks and can recognize subtle changes that can indicate an oncoming crisis. And for many people, a miniature horse is far less intimidating than a dog. A person allergic to dogs may not be allergic to horses, and vice-versa.
As someone with mental health issues, I understand and sympathize with people who use ESAs, but the truth is that most of those animals aren’t providing active support to their owners, which is why they are not considered service animals. I love my cat. He does wonders for my mental health. But it’s passive. He’s not watching me for signs of a crisis and stepping in. He isn’t trained to get help if I am having a crisis.
Most dogs or horses won’t even make it through service training. The requirements are incredibly rigorous. That’s why I have no problem with certified service dog or horse, but please for the sake of your (generic you) pet at the very least, keep your untrained or minimally trained or barely housebroken pet at home. If they must come with you, please keep them in an appropriate carrier for their comfort and safety as well as that of everyone around you. If you love your animal so much that you rely on it for emotional support, maybe try caring as much for its welfare as you expect it to do for yours.