Your question is old, but I think it's interesting, so I'll post a few reasons.

Why we call it pussy.

Because they are furry, down there.

Because at first they are like a strange cat. Snarling, hissing, etc., but once you get their friendship, they roll on their back and suck up all the attention you will give them.

Because some poor country hick couldn't afford a sheep, so used a substitute, and was so satisfied, he ran thru the village, screaming of the encounter causing all the menfolk to grab their women and get some ‘pussy’ themselves.

Because some women have very feline habits. (finicky, oft wonder off to do their own thing ignoring you. Eat only when they feel like it, not at mealtime, and have a tendency to walk at their own pace, again ignoring you (ever tried to walk a cat).

Because women have feline movements. Esp some. Saw one that while filling my prescription moved so cat like and sexy, it was better than any exotic dance. Back in the days before glassed in and over crowded drug stores.

Because of the way they curl up on the couch, like a cat. All tucked in and cozy.

Because if you go down, and do it right, it’s like a cat cleaning itself.