Visual intuition for direct sum vs. tensor product of vector spaces

This is probably not what you are looking for, but my "geometric" picture for tensor product comes from function spaces. Consider $V$ to be the space of functions $f:A\to K$ and $W$ the space of functions $g:B\to K$ for two sets $A, B$ and a field $K$. Then the tensor product of $V$ and $W$ consists of the functions $A\times B \to K$.

If it comes to bases and such, the simplest case is the case of Hilbert spaces. If $(f_i)$ is an orthonormal basis of $V$ and $(g_j)$ is an orthonormal basis of $W$, then an orthonormal basis of $V\otimes W$ is given by the functions $(f_i\otimes g_j)(x,y) = f_i(x)g_j(y)$. If find this picture pretty intuitive and geometric. Even more concrete is the example of $A=B=[0,1]$ where the orthonormal basis in both spaces is the (real or complex) Fourier basis.

As a concrete example: The space of square integrable measurable functions $L^2(\Omega)$ it holds that $$ L^2(\Omega_1)\otimes L^2(\Omega_2) = L^2(\Omega_1\times\Omega_2) $$ and if $(\phi_k)$ and $(\psi_k)$ are orthonormal bases of $L^2(\Omega_1)$ and $L^2(\Omega_2)$, respectively, then the functions $$\Phi_{j,k}(x,y) = \phi_j(x)\psi_k(y)$$ form an orthonormal basis of $L^2(\Omega_1\times\Omega_2)$.