Some rough ideas:
-$\small{End(V\otimes W)\cong (V\otimes W)^*\otimes (V\otimes W)\cong (V^*\otimes V)\otimes (W^*\otimes W) \cong End(V)\otimes End(W)}$ and this remains true as $G$-modules.
So your object is almost $T^2(\mathfrak g):=\mathfrak g \otimes \mathfrak g$ with $\mathfrak g=Lie(G)$. (in fact $(\mathbb K\oplus \mathfrak g)\otimes (\mathbb K\oplus \mathfrak g)\cong \mathbb K\oplus \mathfrak g \oplus \mathfrak g\oplus T^2(\mathfrak g)$ as $G$-modules)

-$T^2(\mathfrak g)=S^2(\mathfrak g)\oplus\Lambda^2(\mathfrak g)$ where elements of $S^2(\mathfrak g)$ are polynomial functions of degree $2$ on $\mathfrak g^*\cong\mathfrak g$, so it is well understood.

-If you want to look at invariants on $T^2(\mathfrak g)$, you will consider $S(T^2(\mathfrak g))\subset T(T^2(\mathfrak g))\subset T(\mathfrak g)$, the tensor algebra on $\mathfrak g$. I think that some people know something in type A for such invariants. Maybe are there some hints in Procesi's paper: http://arxiv.org/abs/1501.05190

-for specific elements, you can consider stabilizers of a pure tensor $x_1\otimes x_2$ with $x_1$ and $x_2$ are semisimple. Then $G^{(x_1\otimes x_2)}=G^{(x_1,x_2)}$ (the second is the simultaneous stabilizer of $x_1$ and $x_2$ in $G$). I can't ensure yet, since I have to leave, but I think that this stabilizer is generically finite, even in this very specific case.

Edit: of course, if $x_1$ an $x_2$ are regular semisimple element lying in different tori, the simultaneous centralizer is trivial.