Is there a name for this map induced by bilinear forms?

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

The correspondence you describe is part of the definition of the tensor product: $V \otimes W$ is defined to have the universal property that for any $U$, we have $$\operatorname{Hom}(V \otimes W, U) = \operatorname{Bil}(V \times W, U).$$ I wouldn't even give it a different name: the bilinear form is the same as the map out of the tensor product.

A bilinear form on $V$ (if non-degenerate) lets you identify $V$ with $V^\star$: $v \mapsto \langle v, \cdot \rangle$

In this case, your map is just a contraction of the identity map on $V^\star \otimes V$ (which, considering our identification, is the same as the identity on $V \otimes V$).

http://en.wikipedia.org/wiki/Tensor_contraction