I don't know a specific name for that, but I would call it associated. I wouldn't call it induced, because the map $\text{Bil}(V) \to (V \otimes V)^*$ is one-to-one, since every linear $T \in (V\otimes V)^*$ induces a bilinear form on $V$ by sending $(v,w) \mapsto T(v\otimes w)$ and this clearly is the inverse.
Kind regards Konstantin
