Yeah, for me this is all about tone and theme and what you’re all after with the game. Joke characters are just fine in some games, and completely derail the whole affair in others. I do think there’s a difference between unusual characters who might be playing against type and characters that are straight up anachronistic in the setting, or built as one-note FUs to the vibe the GM and other players are after.
As for how I’d deal with it? I’d say nope when it was appropriate (if we’re playing a human-focused low fantasy game, sorry, no other races need apply) and “Sure!” if the game was such that dropping a whacky idea into it won’t harm anyone’s fun.
I ran a Tiny Supers game a year or two back, intended to be a four color affair where the heroes wear spandex, have secret identifies, the works. A good friend of mine made a little old lady who could summon and talk to ghosts… and that’s about all. Wasn’t in-keeping at all with the original intention of the game, but I figured a comic book reality could withstand that weirdness… and it did just fine. In fact, those ghost would foretell of terrible things about to happen in obtuse ways, so it ended up being pretty cool.
Set expectations up front. Talk about it. I’ve found that generating characters in a vacuum before session zero isn’t always a great idea. I also choose the systems I run carefully – many of them come with guardrails and methods to “protect” the genre being emulated, and I’m personally a fan of that.
Edit: And forgot one thing… I just have the campaign world and the cast in it react believably, in-keeping with the internal logic (or lack thereof) of the game. I’ve just gone from running The Rad-Hack to Delta Green, and the shifts in presenting the world to the players have been a bit mind-bending. (I’m actually using DG as a refresher to clean all the gonzo out of my brain. I’m a little tired of gonzo at the moment.)