I guess in theory, there is no real difference between pet therapy and sitting next to an animal hanging out. I’ve never heard of anyone charging for pet therapy though. Usually it’s volenteers in a community. they have a dog that is particularly good with people, and then engage in special training for the dog to get them used to the environment, and also to ensure that the dog is calm enough to be welcomed into a hospital, retirement home, etc without causing more trouble than it helps.
the purpose of the pet therapy is usually to bring that extra brain stimulation as Doctor D has said above. that extra brain stimulation is often associated with an increase in pleasurable chemicals released into the body (like when you pet a dog, you feel happier, those are the chemicals working). The pet therapy also gives the patients or retirees a companion to enjoy the company of. Also a long history of emotional connection isn’t required for a person to feel comfortable with a pet like it often is for a human.
the last reason i can think of is that pets, particularly dogs, are incredibly adept at sensing when something is wrong with someone. For example a heart attack or a stroke, a dog can tell when it is happening way before the person even begins to show the symptoms. That is one reason why people who have seizures often have a dog with them, when they are about to have a seizure the dog will go nuts, barking and the person knows they are about to have a seizure and can lie on the ground before they fall to it, and seizure dogs are trained to actually lie on their owner which helps them to not thrash around during the seizure as much. Therapy dogs don’t usually go that far, but any dog, when a person is about to have a seizure, stroke, or heart attack will start barking like crazy and alert people to a problem before it happens. It’s really quite amazing