Could this also mean "you're wearing your cat"? - or, if you don't like the present progressive, "you wear your cat"?
I believe that would be "ti vesti..." from the reflexive vestirsi, to dress oneself.
Exactly. English don't use reflexive verbs but they are quite common in Italian.
"you dress your own cat" is wrong?
Enough already with the dressing or no dressing of your/my dog and cat!
I think the word is "poverini": http://themetapicture.com/these-girls-dont-need-a-hobby-they-need-a-boyfriend/