Let me introduce you to the word “pusillanimous”. It means showing a lack of courage or determination, or being timid.
So when you call somebody a 'pussy' for not having the courage to do something that has _nothing_ to do with felines or vaginas. (As most people commonly seem to think). Being a 'pussy' is short for 'pusillanimous', nothing to do with cats at all…
It can mean, for example, variations on "Don't be so timid", "Stop expressing your fears", and "Take more risks".
Indeed, I have heard "Don't be a bitch" be used in almost the exact same way.