Before 1982 the federated kingdom north of the 48 contiguous states of the U.S.A. was "the Dominion of Canada". In 1982 the basic constitutional law of Canada ceased to be an act of the British parliament, which could be amended only by the British parliament, and came entirely under Canadian control. This step is called "patriation". Did some other word replace "Dominion" after patriation?
The "Dominion" part for the name of Canada fell a bit out of use as early as the 1950's.
Canada Act 1982 and Constitution Act, 1982 changed a bit. But while Queen Elizabeth II is still monarch of Canada and therefore you might call it Kingdom of Canada, the name of the country is now just "Canada", usually.
and other polities with which Canada has official relations as a state consistently use Canada as the only official name, state that Canada has no long-form name, or that the formal name is simply Canada.
While no legal document ever says that the name of the country is anything other than Canada, Dominion and Dominion of Canada remain official titles of the country.
The emphasised "you" is meant just to highlight that under these circumstances it would be unconventional, but in my opinion legitimate, to draw attention to this "oddity" by calling it a kingdom. Not many do that, but outside of diplomatic or political circles, I do not see much reason not to use that term.
The head of state of Canada is called – or rather titled – in Canada:
English: Elizabeth the Second, by the Grace of God of the United Kingdom, Canada and of Her other Realms and Territories Queen, Head of the Commonwealth, Defender of the Faith
French: Elizabeth Deux, par la grâce de Dieu Reine du Royaume-Uni, du Canada et de ses autres royaumes et territoires, Chef du Commonwealth, Défenseur de la Foi
Wikipedia lists Canada as being:
Government Federal parliamentary constitutional monarchy
• Monarch Elizabeth II
• Governor General Julie Payette
• Prime Minister Justin Trudeau