In 2D, using either the angle or its tangent needs a single number. The third alternative is, as others have mentioned, to use a complex number (i.e. the cos and sin couple).

Any of these 3 (angle, tangent of angle and complex number) may be the best choice for a given problem, but for 2D graphics applications I think that using a complex number is more frequently the best. For 3D problems there are 3 corresponding alternatives, using a pair of angles, using a pair of tangents (i.e. coordinate ratios) or using a quaternion.

Specifying directions by the ratio between increments in orthogonal directions, instead of using angular measures, has always been frequent in engineering, since the Antiquity until today.

For something like cursor movement, the ratio between Y pixels and X pixels clearly seems as the most convenient means to describe the direction of movement.