It is very similar to slave and master in BDSM, where one is dominant and the other is submissive. It doesn't always have to be sexual, but a lot of the time it is. Pets usually act more animal like, which is why they are called so and not slaves. It's just another power play, bondage fetish.
So to answer that second question, no it's not like the companionship people have with their cats and dogs.