Since there is no mention of it, I will make one.
The direct product of graphs $G$ and $H$ is defined like this: $V(G \times H) = V(G) \times V(H)$ and $E(G\times H)$ contains only $((g_1, h_1),(g_2, h_2))$ such that $(g_1, g_2) \in E(G)$ and $(h_1, h_2) \in E(H)$.
In your properly edge colored graphs, if one considers all the pairs of monochromatic subgraphs (i.e., maximal subgraphs all of whose edges are the same color) color)---one in $G$ and the other in $H$---and takes their direct product, then one gets the same result as your product.
The direct product is well-studied---see the book by Imrich and Klavzar. There are also other products such as the normal (strong) product, lex product, cartesian product, etc. that might also be of interest.