Canada's national identity has never had anything to do with being subject to anything or anyone. That isn't what the monarchy is about now and it wasn't what the monarchy was about in 1867. Remaining a member of the Commonwealth and maintaining the country's monarchy is less like staying in your parents' basement and more like keeping the surname with which you were born. The monarchy is Canada's surname, and just like how having the last name Johnson does not mean that your entire identity revolves around being the son of someone named John, remaining in the Commonwealth of Nations does not mean that Canada's national identity is grounded in servitude (or in 'Britishness').
I suppose, but she is still technically the head of the state. We still have her sitting in a position of theoretical power. And while born-Canadians don't have to swear an oath to her, all immigrants do (literally swearing servitude). The government's legal cases are still called "The Crown Vs ______". Not to mention Trudeau's trip to England, where he praised the Queen for signing our Charter of Rights in the 1980's. I'd say all that makes her more than just a last name. We're still an active member of the British "family", and still go to all the family reunions.