Is there a graph product that is multiplicative in independence number?
Consider $G = K_2$, for which $\alpha(G)=1$.  The only product defined in terms of subgraphs of $K_4$ and for which $\alpha(G \times G) = 1$, is the strong product (⊠).  This doesn't quite work, since $\alpha(C_5 ⊠ C_5) = 5$, which is not a perfect square.

However, denoting the lexicographic product by $\circ$, there is an easy proof due to Geller and Stahl that $\alpha(G \circ H) = \alpha(G)\alpha(H)$.

So you want the lexicographic product.