A slave is kept against their will. They are forced into labor to produce a profit. You are not profiting off the cat you adopted.
Also, animals have a lot of rights given to them and as a community we are not tolerant when another person in the community does not follow the rules for animal respect. Ironically, those same guidelines do not exist for slaves today in counties around the world and also in the past.
The animal is not as complex as a human. A human can notice their history and where they stand in the world. They also have long term goals and short term goals. They are self aware. Animals do not posses these traits. They are at a primitive level, cognitively. Animals only focus on immediate gratification, such as sleep, food, sex and entertainment, like being patted. Since you provide all of this consistently for the animals, they are interested in sticking around with you.
(The idea you mention above about victims falling in love with their kidnappers is called Stockholm Syndrome.) In order for one to have this, they must be manipulated, meaning they must have a set of rules they independently established BEFORE they got brainwashed and in order for one to get rid of Stockholm Syndrome, they must undergo treatment, since animals do not have the same type of self awareness of humans, they can not be brainwashed or un-brainwashed.