As an outsider I consider Canada and Australia - British Empire. Hard to think otherwise as they have the same king.
Does their king have any more hard powers in Canada or Australia than, say, Taylor Swift or some other rich celebrity?
Kind of! There's nothing on paper that says the King can't just decide that democracy is over and dissolve Parliament. Another example is that the King's representation in Canada, the Governor General, unilaterally gets to decide whether or not to give any passed legislation "royal assent." However in practice, they always do and they otherwise never put their thumb on the scale. Doing so would be a constitutional crisis that would likely end our relationship with the Monarchy more formally and put pen to paper that no, you don't actually have any real kingly powers.
The Governor General has in recent times prorogued Parliament when the Prime Minister asked them to. Ie. "This is politically nasty. Let's hit the pause button and come back when things are better and we're not about to be ejected from power..." And that has been politically controversial. Historically the Governor General just says yes because they want to avoid playing a political role at all (ie. preserving this convention that the Monarchy is really just a decoration of our government).
