I'm not sure but I think the original utility and motivation for this mathematical puzzle is how to represent possible legal moves when programming chess, and this would be evidence that an 8bit unsigned integer is sufficient for the worst case scenario, although you would need some complex kind of encoding mechanism to make the representation terse enough to represent the common moves along with 7 promoted queens in the same 256-moves space.

Practically I think I'll stay with a fixed-length encoding for each of the starting pieces and their movements assuming maximum freedom, while adding a variable length variable in case of promotions.

Although nowadays with OOP and classes and superfast CPUs you probably have entirely variable length encodings, you know, an array of piece objects each with their own legal_moves function. But back in the day, when chess engines were written in C, these things were managed globally with all kinds of hack to save space, not due to space reasons, but to improve locality by reducing cache sizes.

For example, even though the chess board is 8x8, a common trick is to make the board 12x12 to account for knight moves that go off the board (and mark them as ilegal of course.) Which goes to show that even with efficiency as the upmost consideration, a terse representation is not ideal, so I doubt we are going to see 8bit variables to represent moves.