It should just vary based on language. In English it's "English Channel", in French it's "La Manche", in Italian it's "La Manica" etc. etc.

It's not a UK vs France thing, it's just English vs French (vs every other language).

Things get complicated when governments make things "official", though. For example, the Welsh government decided to make the "official" names for some places Welsh, which English speakers have no idea how to pronounce. So the Brecon Beacons is "officially" Bannau Brycheiniog, even in English, apparently.

It still says "Brecon Beacons" for me. Doesn't even include the official name in brackets.

That is due to differences in language.

In Ireland, for example, it's called the English Channel.