::::Your first quotation does not support either of the two disptuted statements ("Canada has been a monarchy since 1534" and "there have been 33 sovereigns of Canada"). While I might dispute that the office of the monarch is the oldest continuous and unbroken institution in Canada—could not it be the Catholic Church?—the second quotation says nothing about 1534 or 33 sovereigns. All it says is that the Crownas an institution has existed in Canada since before 1867, a point not in dispute. Third quotation is not from a reliable source, which is a good thing, since it's baloney. The Canadian Crown has nothing in it of the old French crown and the claims made by the French which were transferred to the English in 1763 could just as well have been made and transferred by a republic as by the French monarchy. The 1583 date seems to contradict our assertion that 1497 is the important one. 
 ::::The statement that "32 French and British kings and queens have reigned over Canada" is not the same as "there have been 33 sovereigns of Canada". The term "sovereign of" has different implications from "reigned over". The Canadian Encyclopedia's statement is not objectionable. I don't think royal.gov.uk is a reliable source for reasons already stated, but its statement seems to trade on ambiguity—not to mention that it is in conflict with the dates of 1497 and 1583 for the start of British monarchical rule. The ambiguity is in the word "monarchy", which could mean a type or system of government, or could mean a country with a monarch as its head of state. Canada (as this article understands it) was not such a country prior to 1867, but the French and British colonies (there were several) were always under a monarchical form of government (with the exception of Cromwell's time). 
 ::::"There may have been two lines before 1763, but its evident neither of those two is considered apart from the present one." You can't consider them apart, because what is Canada now (and what was Canada in 1867) was divided between them in the 16th–18th centuries. It is silly to treat these monarchs, often at war with one another, as being a single monarchy, as this article does.