This proverb traditionally meant to say something about things with just one value (i.e. cats are useless but for their skin), but I read a story once where it made more sense to me. The story used it dually, about a woman subservient to her abusive husband, who obviously felt she was good for just one thing, and that he owned that thing, but as she struggles with her own power, it also came to mean that while you can own a cat’s skin, the essence of a cat will always elude you. The details of the horror story (it was ostensibly a horror story) have mostly faded, but I remember that proverb in that light, a description of something that cannot be owned no matter how you claim it physically.
I love that about cats. They behave independently, totally disregarding how you may suppose you own them. Cat is named after Capote’s Cat, the poor slob without a name, out of my love for this trait. “We belong to nobody, and nobody belongs to us. We don’t even belong to each other.”