A picture is worth a thousand words – so ‘hir’ it is:
We don’t need a mode for that, I think…
Double click on the face is the standard for that… 
It would be useful to double-click on the model face to select the UV island as well.
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Yeah… that’s how it usually works everywhere…
https://media.discordapp.net/attachments/587761956526751784/690665059252174978/2020-03-20_21-54-07.mp4
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If you go to face mode and hover over and island, pressing “L” selects the island. If you want a work around instead of using the island mode when you turn off UV sync mode.
Yes, Blender already has island selection mode, but it’s hidden when the selection sync mode is active, because mesh edit mode doesn’t have UV island mode so it can’t be synced in both directions.
It’s a bit silly limitation though, which could have been handled much better. If you select island in UV editor with sync enabled, then it’s not a big deal since faces making up corresponding island can be easily selected in the viewport using face select mode.
But the other way around is not a big deal either. If someone selects some faces in 3D view while sync is active, I’d simply switch UV editor from island mode to face mode, and selecting the same faces. That way, island button could remain available in sync mode, and UV editor would simply switch itself to face mode if user made face selection in 3D view.