That’s the funny thing about language. How words transform doesn’t always make sense. Virtually nobody who is a native speaker of English uses the word “pussy” to mean “cat/kitten,” anymore. It is almost exclusively used to mean a vulva, and whether it started as meaning a cat or not when used as an insult, I believe it no longer carries that same connotation of “cat” as its primary meaning.

I would suspect, though this is just my guess, that you would be in the minority, today, for thinking that pussy being used as an insult refers to kittens. I agree that makes more logical sense, but language and idioms aren’t always logical.