Basis free description of direct sum-tensor product natural isomorphism
That isomorphism holds for modules as well (for example, using adjunctions), and bases are not necessary. The elements of $(W \oplus Z) \otimes V$ are finite sums of pure tensors, which are of the form $(w+z) \otimes v$. Such a pure tensor is mapped to $w \otimes v + z \otimes v \in (W \otimes V) \oplus (Z \otimes V)$.