Um, Mexico is actually officially Estados Unidos Mexicanos, or, the 'United States of Mexico'. The use of simple 'Mexico' is colloquial and has become normalized.
So, what exactly is a normal name?
The US was founded (and still exists) as a collection of state governments acknowledging and authorizing a central government on the "American Continent."
These United States of America...
There exists a difference between the terms, "Province." and, "State." A province may be a portion of a state in fact; provinces are typically administrative divisions which are under a state. Definitively, provinces would be similar to a district or county (i.e. Pike County, KY; Montgomery County, PA; etc.) A state on the other hand is a sovereign political-geographical entity, synonymous with the term, "Country." as stated by Gadarn. However yes it is correct that this is the reason for which the Union is named the United States of America; indeed the European Union is a fine example, and perhaps the United Kingdom of Great Britain and Northern Ireland may be used as a similar example, although controversial to some, especially with consideration to the approaching Scottish independence referendum. A nation is typically named based upon what their form of government is: the Federal Republic of Germany; the Kingdom of Spain; the Hellenic Republic; etc. Thus states may have "normal" names such as those listed above: the Commonwealth of Kentucky, for example. Also, as stated by others, the name US of America does indeed derive from the Union of the American continent, or North America (some countries recognize N. America and S. America as a single continent.) It cannot be definitively known what the name of the US would have been had it not been established as it was; potentially and likely, it would have had a name which referred to its governmental form (i.e. Republic of..; Kingdom of..) followed by whatever name would have been chosen for itself. I hope that clarifies and answers your question.
Canada actually used to be called the "Dominion of Canada", from about 1871 to 1982. It started moving away from the Dominion part in the 1950s; World War II led to a certain autonomy to the country. Due to the historic ties to Britain, we entered the Great War and the Second World War when they did.
As an interesting side note, Sir John A MacDonald (the first Prime Minister, and one of the leaders of the charge to form the country) actually wanted it to be called the Kingdom of Canada (it was, and still is, governed by a monarch). Leonard Tilley, the Premier of New Brunswick at the time, though, suggested Dominion after reading Psalm 72 (he was a very pious man, and big in the Temperance movement), line 8 of which reads "He shall have dominion also from sea to sea, and from the river unto the ends of the earth."
In any case, Canada used to be "The Dominion of Canada", not just "Canada"; it's only the impact of the world wars, and a general change in government and social policy and ideas, that moved it away from that to just "Canada", with that becoming more official through the Constitution Act of 1982.