First, I think it has to do with some pre-Christian beliefs in Europe about animals as divine spirits, omens, or gods. The word ''monster'' actually means ''warning'' because some people thought that an animal with a missing leg or extra eye or something was a message about the future. That became a superstition. If a ''witch'' looked at animals too much for signs about life or the future, then that could be considered distracting or deviating from the dominant religion of the time, when communities were only supposed to find spiritual fulfillment in angels or saints or God. Not many people knew how to read in medieval times, so even the translations to English weren't read. There can still be animal symbolism in the Bible, I think the original cherubim or seraphim had four animal heads and wheels(???). But if you want to fit into a society like medieval Europe, then it's much easier to separate the ''bad pagans and their animal gods'' from ''good Christians and human-shaped angels and saints''.
So, during the witch hunts, people would say that so-and-so has a ''familiar spirit'' that took the forms of animals, either a spirit animal or a ''demon'' that took over the body of an actual animal. That made it easy and obvious that it was Bad. I don't know why cats in particular would be so discriminated against. Maybe just that cats are more difficult to train, so villagers thought they were untrustworthy even if cats preyed on vermin.
Fast forward hundred of years when the witch hunts are over and most people know that it was just hallucinogenic mold in the bread making people act like jerks. The witch becomes more a cartoon version of what people thought, spread around with rumors and superstitions. In modern times, we keep animals as pets, cats being one of the more popular ones.
If somebody wanted to be a ''witch'' then the obvious place to start would be to imitate the common knowledge look of a witch, with the broom and the cat. Some people who are drawn to witchcraft might also just so happen to like cats, and maybe that works for them.
Animal spirituality is definitely interesting, but when it becomes common knowledge that one must have a pet familiar to be a witch, then yes I agree it loses a lot of nuance, especially when familiars (or spirit guides, or even gods, they're the same to me) can be neither physical nor in the form of animals.