Mathematics also uses tensor to refer to elements of a tensor product of two vector spaces. A tensor product of an n dimensional and an m dimensional space gives an n x m dimensional space. So to the extent that an array is a vector, a higher dimensional array is in a tensor product of vector spaces, and so we call it a tensor. Or, there's a correspondence between k-dimensional arrays an a tensor product of k vector spaces determined by a choice of bases of the vector spaces. And we're quite comfortable picking a "standard" basis for R^k, so we conflate the two ideas.

Now the part that always bugged me was that, at least in C++, a vector is an array that can change length, which is a very non-vectorial thing to do.