When I became a Canadian citizen, around a decade ago, I pledged fealty to Her Majesty Elizabeth the Second, Queen of Canada.

So, Canada (and probably Australia too) is a kingdom different from the United Kingdom, that happens to have the same monarch. Different crowns, same person under it. Given that the British monarch only retains a ceremonial role in every country that they “rules”, having the same one doesn’t make a lot of differences from having our separate one.