To some people, national identity is very important and cannot be mixed. I am England born and bred and I know of others who are the same who are proud of that fact and would never consider themselves to be British but English. There are Welsh people who identify themselves as Welsh and some Scottish people are Scottish.
Canada is not part of America. The United States is formed of 50 states south of the US/Canadian border and South America is south of the US.
Generally speaking it would be correct, just like English, Welsh and Scottish people can be classed as British, but nationality speaking, Americans are from the US, people from Wales are Welsh and people from Scotland are Scottish. I have spoken to people from Puerto Rico who refer to themselves as Puerto Rican.
You need to be careful. Try not to mix nationals with other nationalities.
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In the present climate in the UK there are people who want to break the union between UK and Europe on a political basis. We are European geographically and always will be but the UK does not form Europe. It is part of Europe.
The UK is "The United Kingdom of Great Britain and Northern Ireland" and Great Britain (or Britain) is England, Wales and Scotland.