Part of the reason that humans have such a close relationship with dogs and cats was a practical one. We domesticated dogs first because we need protection from other predators as we traveled and hunted for food. Then as learned husbandry, they became our partners in protecting the herds and rounding them up to either change pastures or to sell. On some level, people still get dogs because they like the added protection on top of the companionship.
Cats were domesticated later around the time where agriculture got popular. Because the crops attracted animals like bunnies, rats, and snakes the cats came. They were seen as beneficial because they ate the animals that would ruin crops and get into our storages and eat our food. Many cultures admire the calmness of the cat and its quiet companionship.
Rabbits are indeed very cute as pets but they need to dig to be happy. And historically speaking, they were considered food or pests until fairly recently. Rabbits eat crops and burrow in your backyard which could affect your ancestor's survival through the winter.
Then wild animals such as monkeys, snakes, most reptiles, and other wild animals usually fall into two categories. They can either kill you, and you need to be a certain personality type to get off on having a dangerous animal. Or they are just weird enough that it's hard to get their food or take care of them, which can be unappealing. Plus, most of veterinarians are trained to take care of dogs and cats but have a harder time when dealing with a pet scorpion. So part of the reason dogs and cats are so popular is because they have a functional place in most societies while the other pets are more because we just like the animal.
