If you're talking about sports teams, they're basically arbitrary. Many independent countries are split into multiple "national" teams for some sports/competitions, and sometimes multiple independent countries form a combined team. Most sports have some kind of international governing body that decides how this works, though they usually take into account the preferences of local organisations and individuals.

> even though they are part of a larger country the UK, that would not also make something like a US State a country?

The thing is, from the British perspective there is essentially the same issue. The UK is a single "state", but the US is split into multiple "states". Each country/state has its own unique political system with its own terminology, and whenever we try and talk comparatively between countries things can get a little bit confusing and vague. And it's not just the terminology - Wales is a different kind of thing from, say, Michigan, not just in the sense of what we call it, but also in the sense of how it fits into the politics and culture of the country/state of which it is part.