The way to think of it is thusly; rather than having a revolution and severing the constitutional ties with the British Crown, Canada evolved out of Britain in stages, (we were actually about to have a revolution and join the Americans when James Madison invaded and in the process created a "Canada" as a reaction) but just because Canada is not the United Kingdom, does not mean that it is not the British Crown, this is still the British Crown here, and the Constitution flows from the Crown not the government, the Crown simply exists above the level of national governance, to wit. the government is not actually all that significant in the grand scheme of things, or at least things important as per going over the top into no mans land.
My country is the House of Windsor, the Queen my Sovereign and Commander-in-Chief, never swore an oath to "Canada", tho I am bound to Her Majesty by a blood oath unto death as necessary. England, Scotland, Wales, Ulster, Canada, Australia, New Zealand, etc, not the same government, but the House of Windsor all, and all under the British constitution, Canada being the only one which added a Canadian Bill of Rights on top, Australia, New Zealand for example are still governed by the British Constitution without a domestic Bill of Rights of their own.
More British than the British, more American than the Americans. Suck it, Aussies.