This question is specifically in reference to
Dimension of spaces of bi/linear maps
but since it is a new question only indirectly related to that post (insofar as notation is concerned), I'll ask it as a new question. I am somewhat of a newcomer to multilinear algebra, so forgive the naïveté.
Since $V\times V$ (or $U\times V$ for that matter) is a product space, why would $\operatorname{Hom}_{\mathbb{F}}(V\times V,W)$--where $W$ may but is not required to be the ground field $\mathbb{F}$--be interpreted as anything other than the space of bilinear maps from $V\times V$ to $W$? That is, an element of $V\times V$ is a pair of vectors $(\boldsymbol{u},\boldsymbol{v})$ so a map that acts on such elements is naturally bilinear, right?
I thought one of the primary purposes of the tensor product was to "convert" multilinear maps to linear maps so that the full machinery of linear algebra can be brought to bear. I get that $\operatorname{Hom}_{\mathbb{F}} (V\otimes V,W)$ is a space of linear maps, but why wouldn't
$$\mathrm{Hom}_{\mathbb{F}}(V\times V,W)$$ be the space of bilinear maps,
$$\mathrm{Hom}_{\mathbb{F}}(V\times V\times V,W)$$ be the space of trilinear maps, and so forth? I have no particluar objection to using notation like
$$\operatorname{Bilin}_{\mathbb{F}}(V\times V,W),$$ but is it necessary? It will become cumbersome once you start working with larger product spaces.
I am probably missing something obvious...
You say $V \times V$ "is" a product space, but it is also a vector space (with dimension twice that of $V$).
A mapping $V \times V \to W$ could be bilinear or it could be linear or it could be neither. The "Hom" notation in ${\rm Hom}(V \times V, W)$ means linear maps $V \times V \to W$ with $V \times V$ treated as a vector space, which are not the same thing as bilinear maps.
Example. For two linear maps $A, B \colon V \to W$, the map $L \colon V \times V \to W$ where $L(u,v) = A(u) + B(v)$ is linear but is not bilinear (except in silly cases like where $V = 0$). So this $L$ has nothing to do with linear maps $V \otimes V \to W$.
Example. The addition map $\mathbf R \times \mathbf R \to \mathbf R$ is linear but not bilinear while the multiplication map $\mathbf R \times \mathbf R \to \mathbf R$ is bilinear but not linear.