Earlier quoted context omitted.
Of course not (which I indicated fairly clearly too). To make a fair comparison you want to consider what kind of a feeling or connotation "You're a dog" vs "You're a cat" invokes. Variations on the sentence are okay too, like, say "You're such a dog" or "Come here dog". I didn't contest 'gata', gender specificity does make it non-ideal, but I'll take it.
"You're such a pussy" or "Don't be a pussy" can definitely be used in a derogatory fashion, especially in groups in which social status is dominated by masculinity. 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.
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…