I'm certain there is no situation like that. It's just a dead part of the language. Most native speakers don't even know how to use it properly when imitating old-timey speech.

Quakers use "thee/thou" sometimes, but only because the movement has been around since just before the end of the shift to "you" for everything, and it's fossilised in as a result. There's a few weird British Isles dialects that preserve it too, but they're not widespread.

It's archaic. I can't really imagine a situation in which we'd use "thou" today for formality reasons. If you say "thou" , you're pretending to be someone from hundreds of years ago or you're quoting the King James Bible or something that is hundreds of years old.

I think a more-reasonable division between formality and informality would be whether or not one uses a title like "sir" today.