Consider Napkin Problem 13D (a):
Let $V$ and $W$ be finite-dimensional inner product spaces over $k$, where $k$ is either $\mathbb{R}$ or $\mathbb{C}$. Find a canonical way to make $V \otimes_k W$ into an inner product space too.
I approached this by working on special cases. Let $e_1, \dots , e_n$ and $f_1, \dots , f_m$ are orthonormal bases for $V$ and $W$, respectively.
I think we want $\langle e_1 \otimes f_1, e_1 \otimes f_1 \rangle_{V \otimes W} = 1$, since it feels like $\|e_1\| = \|f_1\| = 1$ should imply that $\| e_1 \otimes f_1 \|$. We also want $\langle e_1 \otimes f_1, e_2 \otimes f_2 \rangle_{V \otimes W} = 0$, since it feels like the fact that $e_1$ and $e_2$ are orthogonal and $f_1$ and $f_2$ are orthogonal should imply that $e_1 \otimes f_1$ and $e_2 \otimes f_2$ are orthogonal.
So for any $v_1, v_2 \in V$ and $w_1, w_2 \in W$, the inner form $\langle v_1 \otimes w_1, v_2 \otimes w_2 \rangle_{V \otimes W} := \langle v_1, v_2 \rangle_V \langle w_1, w_2 \rangle_W$ seems like a good choice, and it ended up being the intended inner form.
But I'm still very uncomfortable with the intuition behind this; is there a more solid way to understand why we take the product of the individual inner forms to get the inner form of the tensor product?