The statement is false. The question is: do you have angular momentum? If you leave the water to rest for an hour usually there is no spinning. If you give a little swirl to the water in the sink before you let it out, you can probably force it to start spinning.
Bathtubs have larger dimensions, you a smaller angular momentum of the water is enough. Oh and forget about the Coriolis force. Australian bathtubs are no different than European ones.