What classification do these areas fall under?

They are all countries, yes.

“Country” may have multiple meanings here.

England, Scotland, Wales, and Northern Ireland make up the country of the United Kingdom, and The Republic of Ireland is another country.

But then England, Scotland, Wales, and Northern Ireland are called countries of the UK.

The link has some explanation of the politics.

@dxs. Thank you. The UK and Ireland are usually listed as two countries that encompass all five, thus the confusion.

They also make up the British Isles.

The British Commonwealth for some of them