Question

does it make a sense to call a physical quantity a vector, when its magnitude is zero

Answer 1

When work with vectors using algebra, we add them to and subtract them from other vectors.  

When we subtract  one vector from another of same value and direction, we get a vector which as a zero size.  However, to maintain that the result of operations on two vectors is a vector, we indicate that as the zero vector.  It is for the convenience and consistency of the algebraic operations.

similarly, when a cross product is taken for two vectors which are parallel, the resultant needs to b e denoted by a vector/  That we call as zero vector.
the direction can be arbitrary bu the magnitude is 0.