Some more remarks, from top to bottom:

void Board::set(const size_t where, Square s) {
    if (where >= board_size)
        return;
    squares[where] = s;
}

Instead of returning, you should throw an exception since trying to modify the board outside of its borders is clearly a programming mistake on a 10x12 board. Same for Board::get.

I would inline the two constructors for Chess::Square.

Having const size_t parameters is not useful since size_t variables are always passed by value. At least in the declarations the const should be omitted. In the implementation it's acceptable, though.

In init_classic_board, the code for placing the pieces is so complicated (first place a queen, then replace that with a king) that you could also just have 16 equal lines, each for one piece. Or a little helper function taking a color and a list of 8 pieces, like this

place_pieces(91, Color::black,
    Piece::rook, Piece::knight, Piece::bishop, Piece::queen,
    Piece::king, Piece::bishop, Piece::knight, Piece::rook);
fill_row(81, Color::black, Piece::pawn);
fill_row(71, Color::none, Piece::none);
fill_row(61, Color::none, Piece::none);
fill_row(51, Color::none, Piece::none);
fill_row(41, Color::none, Piece::none);
fill_row(31, Color::black, Piece::pawn);
place_pieces(21, Color::white,
    Piece::rook, Piece::knight, Piece::bishop, Piece::queen,
    Piece::king, Piece::bishop, Piece::knight, Piece::rook);

Regarding the braces: whenever an inner for loop needs curly braces, I also put braces around the outer for loop. I only omit the braces for one-liners and block statements with a one-liner body statement.

Below the case Piece::rook_castle:, the if clause has wrong indentation. You should let your IDE format the code for you. This would also make the space between ){ consistent.

Before the case Piece::bishop:, there is a // fallthrough comment missing.

After the default:, the break has wrong indentation.

In read_move, the cin>>in must be enclosed in an if clause.