To be satisfyingly pedantic, my King isn’t the same King. Mine is the King of Canada. It just happens to be the same person as the King of England and the King of Australia ;)
Some friends and I started playing Crusader Kings 2 recently. If anyone wants a thorough education in the weirdness of feudal monarchy, I highly recommend it, half the game is manipulating weird inheritance laws/scenarios to grow your holdings.
Still it skips over weirdness like the same person being both a sovereign king and vassal of another king. See for example King Henry II of England and Duke of Normandy etc.
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Charles III, by the Grace of God, King of the United Kingdom of Great Britain and Northern Ireland and of His other Realms and Territories, Head of the Commonwealth, Defender of the Faith.
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So in my head canon, he is the King of the UK and Canada … the same person and the same office. Ie there is no King of Canada officially - the title is always King of UK (first) and of other places as well … in short whilst Canada has a King, there is not a title “King Of Canada” that he can hold as well as holding “king of UK”
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Queen Elizabeth II was the first of Canada's sovereigns to be proclaimed separately as Queen of Canada in 1953, when a Canadian law, the Royal Style and Titles Act, formally conferred upon her the title of "Queen of Canada". The proclamation reaffirmed the monarch’s role in Canada as independent of the monarch’s role in the United Kingdom and other Commonwealth realms.
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The other realms the British King has are the Crown dependencies, eg Isle of Man. Australia is the odd one out in naming the UK as one of the other realms. Your head canon was true, before the independence of the Commonwealth countries.
Jamaica has the monarchy. Jamaican forces were part of the US-led coalition that invaded Grenada in 1983, after the communists seized power. The communists found it politically expedient to maintain the structure of parliamentary government, and so the head of state of Grenada also remained nominally, Elizabeth II.
If you're wondering how it's relevant: the titles can end up separating again. The most straightforward way is if the territories have different rules for who inherits the crown on death.
That page correctly describes the different entities that are all called ‘Canada’. It’s not wrong. The one that has its own monarch - the King/Queen of Canada - is the modern one, established in 1984.
Previous to that, the UK’s monarch had dominion over Canada. In 1984 the roles were made distinct.
There is an even more pedantic objection to your claim: there is no such person as the “King of England”; this is like saying Donald Trump is the President of California.
Right?! Listen, I know this sounds like a conspiracy… but I think the monarchy has historically been a mechanism used to centralize tremendous global power under a single banner. We ought to be mindful of that and, perhaps, construct checks and balances against this concentrated power.