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).
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.