I think we should consider dogs and cats separately.
Dogs and men have been together for thousands of years. It's a kind of symbiotic co-existence. Dogs help men to hunt and dogs get to eat what the men cannot finish. After living together with men for so long, I wouldn't be surprised if dogs consider men to be just a superior canine breed. Definitely a lot of men consider dogs to be better than human companions.
Why are dogs so popular as pets? I think one of the main reasons is that a dog is very sincere in its love for its owner. It does not cheat and lie to gain the affection of its owner. It accepts its owner as-is where-is, with no pre-conditions. Not many human beings can beat a dog at that.
Cats are quite a different issue. In many situations, cats come and go as they wish. It's not so much a question of people adopting cats as cats adopting people. Many cat lovers would happily admit to being owned by their pets.
As for the other animals, there is one that I am particularly interested in. I do think that monkeys can be domesticated. Where I used to live, monkeys, specifically the Pig-tailed Macaques, are trained to pluck coconuts. However they are all taken from the jungle, as babies, torn from the clasping, dying arms of their mothers who were shot dead to get the babies. It's a very tragic thing.
What I envisage is a monkey-rearing program where monkeys can be bred and domesticated. I believe it would take only three generations of domesticated breeding to get monkeys which are tame enough to train to do work. For starters, just imagine having a trained monkey to pick up things around the house. Imagine a team of these little fellows helping to bring in the harvest:
I hope I can make this dream come true before my game is over on this planet.