Q:

Why the Coriolis force cannot be a vector in the usual sense?

Why the Coriolis force cannot be a vector in the usual sense?

A:

The Coriolis force on a mass point with mass $m$  and velocity $\vec v$ in a rotating frame with vectorial angular velocity $\vec \omega$ is given by the cross product $$\vec F_C=2m\vec v\times \vec \omega$$ The cross product of two (polar) vectors is a pseudo-vector or axial vector because it transforms like a vector under space rotations but changes direction upon space reflections. A normal, polar vector stays the same.
Note: This reasoning holds, as mentioned, when in the rotating frame both $\vec v$ and $\vec \omega$ are polar vectors. When $\vec \omega$ is an axial vector in the rotating frame, as pointed out in the comment below, and $\vec v$ is polar, then $\vec F_C$ is not a pseudo-vector. A similar case is the Lorentz force $\vec F_L=q\vec v \times \vec B$, where $\vec B$ is a pseudo-vector, $\vec v$ is polar and thus the Lorentz force is polar. Thus a doubt of the assertion made in the question that "the Coriolis force cannot be a vector in the usual sense" is warranted. 

