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'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.