The two diagonals also intersect at right angles. So "orthogonal" is not really a good abbreviation for "horizontal or vertical".
If you want to chase that rabbit, then which two diagonals? Diagonal means either "line joining two (non-adjacent) corners" or "at an angle".
And with the tiles being rectangular rather than square, the primary diagonals do not meet at right angles.
There is obvious value to having a term for "the directions a chess rook can move" and a term for "the directions a chess bishop can move", with "diagonal" being widely used for the latter, and "orthogonal" for the former.