The Queen only has any power at all when she is actually on Canadian soil, otherwise all of the royal duties are carried out by the Governor General (she sits on a throne in Ottawa). Also, there is something wrong with that statistic posted on the site; it has Indence from the UK listed as being July 1st 1867, that is incorrect, that is the date that the already independent Dominions confederated. For example, Nova Scotia got its governing independence from the UK in 1758, and Prince Edward Island in 1773. In 1841 they received full economic independence. All that happened in 1867 was that they confederated into a larger country, not even all of them did either, most of them didn't sign on until later. It is kind of like the US declaration of independence, most people didn't sign it until August, only 2 people actually signed it on July 4th.
Anyways, Canada is a unique country, the only province where people will identify themselves as Canadians is Ontario. In all of the other provinces people generally identify themselves by their province name as well as Canadian. It's actually quite funny too, looking back at some old documents it can be seen that the people of Prince Edward Island at least were actually very unhappy with joining the confederation (nearly furious). There was actually a conspiracy involved back in the 1860's and 70's that involved the British and Ontario. Prince Edward Island had refused to join confederation because during the time the Island was quite prosperous. 99% of the people did not want to join onto this Confederation. In fact, during this time, the Island was at least as close to the US as it was to the other Canadian provinces, maybe even closer. This caused a bit of a stir, the British Crown was worried that Prince Edward Island might actually elect to join the United States, the US had been sending up officials to attempt to do this. In fact, the Island had actually negotiated trade deals with the US. Truth be told, the Island only wanted its independence for itself and did not wish to be a part of any union. Yet, there were people in the government who had a lot of friends among the high up Canadians; so then they delivered to the Island people the myth of progress, that with a railroad, the already prosperous island would begin booming. So they built the railroad, but lied about its expenses, in fact it cost several times more than what they had said it was going to cost. Guess who was there to come along and solve the Islands problems? Good old John A Macdonald and the British (who wanted PEI to confederate with Canada). They not only agreed to pay for the railway in exchange of allowing them into Confederation, but also agreed to grant the Island $800,000, plus $50 per person, and to pay for a physical link between the Island and the mainland (which they never did, in full at least, they only payed for a portion of the billion dollar bridge); so yeah, you can tell a lot of negotiation went on to get the province to join.
There are a lot of really old buildings here (some of the University buildings are over 150 years old), there's a lot of great history written down and stored in this place. With all the Canadian flags around, no one even seems to remember that 150 years ago Prince Edward Islanders hated Canadians and only wanted to be independent (yes there was even violence that required a small military force to put down). All of the people talked about in the official history books are those who were pro-Confederation, none of the ones on the losing side (who were much more representative of the people) are even mentioned, or if they are mentioned it is only one or two times with no significant information. The only way to find the true history of the Island is to look back at very old texts and documents.