We have the following "internal" definition of the direct sum:
A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$.
(Of course the same definition works for abelian groups, etc.)
Is there anything analogous for the tensor product of vector spaces? The situation is clearly different since $V,W$ aren't canonically embedded in $V \otimes W$. On the other hand we do have the canonical cone of simple tensors, maybe this could be used somehow?
I don't have a complete answer, but I think I can add some information. In the spirit of you last comment, the way toweards characterizing the cone of simple tensors is looking at maximal linear subspaces contained in it. Each line in $V$ determins a linear subspace of dimension $dim(W)$ contained in the cone and vice versa. These two families are exactly the maximal subpsaces contained in the cone and they fill up the whole cone. I think basically the dimensions of the families togethter with the dimensions of the subspaces in each faimily pin down the cone uniquely up to isomorphism.
This point of view seems not to be too common in linear algebra, but it is used in parts of algebraic and differential geometry. For example if you look at the projectivisation of the cone, it forms an algebraic variety known as the Segre-variety.