This page addresses an annoying controversy of English usage that crops up from time to time.
I live in the country whose official name is "The Dominion of Canada". That means that I live in the country known as Canada, the form of government of which is a Dominion, specifically a self-governing Dominion within the British Commonwealth of Nations.
Similarly, the official name of France is "La République Française", literally, The French Republic; so its form of government is a Republic.
The official name of the country to the south of my country is "The United States of America". So, obviously, that country is America, and, indeed, its people often refer to it as such, and its form of government is a United States.
Except that this isn't obvious to everyone.
It is sometimes claimed that the country to Canada's south should be properly referred to as "The United States", because America is a larger entity, of which Canada, Mexico, Brazil, Argentina, and several other countries are part.
And, indeed, if we were speaking French, they would have a point: the United States of America is normally referred to in French as "Les Etâts-Unis", and "l'Amerique" refers to the combined entity consisting of North, Central, and South America, which in English is called "The Americas".
But in English, America is one thing, and the Americas is another thing.
To explain why, a number of examples and analogies need to be drawn.
First, let us recall that the three major commercial television networks in the USA are:
Since ABC is the American Broadcasting Company, NBC, which was also nation-wide in scope, simply called itself the National Broadcasting Company to have a different name with a similar meaning.
And then CBS did the same thing; here, like Columbia Pictures, it used Columbia as a term embracing the New World as a substitute for America.
My home country of Canada includes a province by the name of "British Columbia"; should the Huron decide they wanted their name back (Canada is derived from the Huron - or Huron-Iroquois - word kanata), this term could as easily be considered applicable to the country as a whole. Before Canada's Constitution was patriated, our Constitution was the British North America Act
And in South America, there is a country with the official name of the República de Colombia.
While it therefore calls itself Columbia, no one, to my knowledge, has objected to their use of that name because, for example, British Columbia is not included in its territory.
From 1824 onwards, the official name of Mexico was the Estados Unidos Mexicanos, but officially its translation into English is the United Mexican States.
Between 1889 and 1968, the official name of Brazil was the Estados Unidos do Brasil, literally The United States of Brazil.
So this demonstrates that being a United States is to have a form of government which may be applied to countries other than America; it may be applied to Brazil and Mexico as well.
It would seem like this ought to settle the matter. If the Republic of Colombia can be Colombia, and the United States of Brazil can be Brazil, then the United States of America can be America, and calling it that is not a mistake. America is a country, and the Americas are a pair of continents, with the isthmus of Central America lying between them.
Of course, though, if North America, Central America, and South America are the Americas, then it follows that each of them is an America.
So North America is an America, but America is only part of North America. I can see why some people might find it confusing, as it seems to be true that there is only one America, but there are also three Americas. But none of the latter are America, each one is instead an America. So the word "America" has two meanings; it's the proper name of a country, and it's a word (not a name) meaning a conventional New World continent (the "conventional" part is needed to include Central America).