> There is a chess variant like this - Crazy house.

Interesting, I've always heard it referred to as "Bughouse" or "Siamese chess".

https://en.wikipedia.org/wiki/Bughouse_chess

Bughouse is the 4 player, 2 boards variant. You capture your opponent's pieces and give them to your team mate who can then drop them on their board. That way there is no problem with the piece being the wrong color to drop.

Shogi solves that by having the pieces be flat with the two colors on different sides, afaik.

Crazyhouse is online only, so it has no problem switching the color of the captured piece.

the two-sidedness of Shogi pieces is to deal with promotions, not control.

the pieces have a pointy end, which indicates control; you just rotate the piece so it's pointing the correct way when you drop it. when promoted pieces are captured, they also do lose their promoted status.

you can indeed DIY an OTB crazyhouse set with either flippable or directional pieces (here, you can use a marker to mark 180-rotational symmetric chess pieces so they have a "front") though, or even just use a Shogi set to do so with a piece mapping (ignoring the promoted side).

It does seem like it'd be simple to have a second chess set to play Crazyhouse in person, considering you need two sets to play bughouse just without 4 players

Bughouse is the 4 player version played on 2 boards. A lot easier to play with conventional physical boards.