The U.K. is a country. If you mean a country is something that passes the tests for statehood. It’s as much a country as Germany or the Netherlands or the US. It doesn’t have to answer to anyone else like the EU does.
England/Scotland/Wales are countries if you mean the broader definition of country, but they cannot make laws on their own as they are subordinate to the U.K. so don’t pass the tests for statehood. They’re closer to the States in the US, except England doesn’t even have a government so in many ways is less of a country than say Texas.
The problem is the definition of the word country, dictionary says:
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No government: not a country. So England fails that definition.
The EU is nothing like the U.K. because the EU can’t pass laws, it can only request that it’s members do so, it’s more a collection of trade deals and treaties. Like the WTO.