In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to BaajSkandalisVaes. A pair $(G_{1}, G_{2})$ is called a matched pair of l.c. groups if there exists a l.c. group $G$ such that, $G_{1}$, $G_{2}$ are closed subgroups of $G$, intersection of $G_{1}$ and $G_{2}$ is trivial inside $G$ and the complement of $G_{1}G_{2}$ is of measure zero in $G$. Does there exists any notion which replace the last condition by some topological criterion?

I suppose you could ask that the complement of $G_1G_2$ is nowhere dense, or more generally a meagre set. But whether this notion is appropriate or not really depends on what application you have in mind. Also, unless I am missing something, isn't every pair of locally compact groups is a matched pair? Just take $G = G_1 \times G_2$. 

