Το κουτάβι δεν είναι γάτα is more natural.
αγγλισμός. Anclisised phrase
Neither my parents nor my in-laws have ever used the word κουτάβι for "puppy". My mother said the word was used as an insult when someone did something dumb.
I have never heard of such usage. That's probably κουτός (which means silly), not κουτάβι.
Do your parents live in Greece?