To be a vector it is not only necessary to have both magnitude and direction, but it must also obey vector laws of addition. Yes, there are quantities having magnitude and direction but are not vectors as they follow simple algebra laws.

For example, Electric current has both magnitude and direction but it is not a vector quantity as it doesn’t follow laws of vector addition because electric current is given by I = V/R, both V and R are scalars and they don’t give a vector quantity hence electric current follows simple algebraic laws. A similar explanation can also be given to fluid flow in a pipe (Q).