Looks like the French name for the English Channel is La Manche (The Sleeve) and many other countries follow this name. Whatever you think of renaming things, however, the policy of giving all the names for a place was a good one.
Same thing basically with the English Channel. It's just the English Channel. I don't want to see a bunch of different names for it. If the French want Google to show it as La Manche for French people that makes sense.
But I'd say that it's probably not a big deal anyway because how often are you going to run into such a scenario, and even if you do how would Google/whoever capture a lot of informal names for things? A little bit of confusion is fine too anyway.