How can I turn into a cat?

Every power-up has a priority level. Picking up something of equal or higher priority will activate it, but picking up something of lower priority will put it into storage instead. You'll have to forcibly remove your current power-up somehow (e.g. get hit) in order to apply one of a lower priority.

To expand on Toomai's answer, there is another layer to the onion of the power-ups in this game - The Golden Leaf (white Tanooki suit). If you have the white Tanooki Suit on any character, no other power-ups will 'stick'. This is as-designed since that power-up basically makes you invincible and only shows up when you are having trouble with a level.