To precisely the extent that a vector is an array, we have also that a tensor is a multidimensional array. A multidimensional array-of-scalars type is precisely the tensor product of the appropriate one-dimensional array-of-scalars types, in the standard math sense of "tensor product": linear maps out of the former are in natural correspondence with multilinear maps out of the latter.

The only things I see to complain about are that vector spaces don't have to come with a choice of ordered finite basis and that arrays don't have to be of scalars from a field. But if you're already happy with the conflation of "vector" and "array", nothing new is happening in then speaking of multidimensional arrays as "tensors"; it's just continuing the same conflation.