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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.
I can’t see how this would work. The “subject to the jurisdiction thereof” part refers to the children born in the US, not their parents. But don’t quote me on this, I’m not a lawyer.