A lot of people think owning animals like monkeys is cruel and it's illegal to adopt them in many countries. Part of the reason it seems to be frowned upon is that baby monkeys are often taken from their mothers in order to satisfy the demand for pet monkeys. Don't get me wrong who wouldn't want a pet monkey? I'm just not sure it's ethical.
Whereas cats and dogs have been domesticated for thousands of years and have a long history with humans in all parts of the world.
As for smaller animals such as rabbits I think they aren't as popular as they don't spend as much time with their owners as cats or dogs. It's more common for people to walk their dog, hug their cat and sit with them while watching television. Some people have trained their rabbit to sit with them but many rabbits don't like to be held.
Dogs are probably the most popular animal as they will interact with humans to a higher degree than other animals and this is what a lot of people want.
