I expect that if I were offered a veto against a stronger player, I would not be skilled enough to spot which move to veto, and would probably end up hoarding my veto, like in video games where you have a great-but-rare ability that you keep perpetually in reserve.

(“Too Awesome to Use” on TV Tropes. Link omitted - you’re welcome).

But then, I’m a terrible chess player.

Nah, you’d spot one pretty fast when you blundered and they went to take advantage of it. Instead you’d more likely have the opposite problem where you’d veto after a blunder but still be at such a huge disadvantage that it wouldn’t matter much.

It would be pretty neat between players of similar skill level though, then I could see the hoarding taking place.

True, and I can see some fun mind-games where a player might try baiting an opponent into wasting their veto on an apparently-strong move, or by intentionally playing a weaker move that still somehow looks strong but actually masks a now-unvetoable killer move…

Maybe for players under say 1800 elo online, but for players above that this won't work -- "bluffing" isn't really a thing until you're at the very very highest levels of chess, and even then the bluffs are only during the openings and if they call your bluff you are only worse by 0.1-0.4 at the most.

But it's better than in the video game, since the mere threat of a veto restricts your opponent at every move.

As the saying goes, "the threat is stronger than the execution".

I'm also terrible; I'd use it when I inadvertently gave my queen away.

I think if this was played at a GM level, games would be dreadfully boring, for one simple reason: the first player that ever allows a winning threat with only one defence, will lose the game.

This will lead to extremely cagey games where no one ever dares make the game sharp and imbalanced.

Ok, I take your protected queen with my queen and veto you taking mine on next move.

<record scratch>

<Zach Braff voice> How did we get here?

I wonder how well this generalizes to other abstract strategy games like go or checkers.

Is there a known family of "functors" for games like this, e.g. veto or having one opportunity to swap positions with your opponent etc.? It would be cool to see what you could say about the rule modification in a general sense before applying to a particular game.

There is some literature on this, yes. I don't know quite how general it gets.

See for example several books by Elwyn Berlekamp.

One outcome of this work was Berlekamp (IIRC) solving a small class of endgame problem that has eluded professional (full-time) go players for literally hundreds of years.

Is it stalemate or a loss if an opponent vetos your only legal move?

A loss if you're in check; a stalemate otherwise. I.e. same as if that move was considered illegal.