No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?

Question

In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, namely $\hat{b}(v\otimes w)=b(v,w)$. But they claim that this association is not canonical. I don't understand why, since this seems like the obvious choice for the correspondence, no?