One thing I'd like to point out regarding the use of the word pussy to refer to somebody who is weak is that it shares it's origin more with the feline meaning than the female genitals meaning.
And here's Merriam-Webster to back up my assertion a little bit. https://www.merriam-webster.com/dictionary/pussycat
So, yeah, I'm still going to call someone a pussy if they are acting like a pussy.
And I'd like to add to this...vaginas don't act any kind of way because they are a piece of anatomy, so in my mind calling someone a pussy or saying they are acting like a pussy will always evoke a cat and not a vagina.
I agree. I've always associated this term with pussycat.