Depending on how much you love your cat you may or may not consider it a natural person. For the purposes of EVE Online's EULA, cats are not individual, natural persons and are ineligible for EVE Online accounts. Assuming the cat was an adult and a natural person (theories on how to prove that would be entertaining) it could sign up for an account. It could not be permitted access to any account registered in your name, obvious to anyone that has methodically memorized the entire content of the EULA through long, lonesome and caffeine addled night shifts.

Indeed, you would not want to find yourself logging in and all your ISK and precious assets sold to unscrupulous individuals for a dimebag of catnip, which happens all too often. Don't account share people, even with your pets. Fluffy got his eyes on your Nyx!