I can't think of any reason as to why one would have a directional derivative using a vector that's not of unit length. It would always "mess up" the derivative by scaling it by the magnitude of the vector, would it not?
I tried searching online but can't seem to find any particular purpose.
The set of all directional derivation operators $\partial_v\vert_x$ at a fixed point $x$ form a vector space. This in itself is a nice theorem, but there is a deeper reason why this is interesting.
If you consider more general spaces (namely manifolds) than open subsets of $\mathbb{R}$, there is no notion of unit vectors anymore. Directional derivations, however, do make sense. The set of directional derivation operators still form a vector space (called the tangent space at $x$).
TL;DR: In more general theory you can't get away with restricting yourself to unit vectors.