I looked it up, and you are kinda right in the theory of it, and I was wrong in the practice of it. The practical power lies with the Cabinet.
From Wikipedia:
"A declaration of war by Canada [...] is an exercise of the royal prerogative on the constitutional advice of the ministers of the Crown in Cabinet and does not require the direct approval of the Parliament of Canada, though such can be sought by the government."
Americans had 200 year dramatic and violent head start and everyone has figured that out by now. The details of Canada’s status are understandably less well known.
So Elizabeth was a British monarch on Canadian money, and Canadian monarch on Canadian money, uniquely.
Of course she is no longer put on newly made money.