Partitioning a set of 2d polygons into intersection-connected subsets
What you want is to find the connected components of the intersection graph of the polygons, right? It would help to be more clear about what counts as an intersection: do the boundaries have to cross or does one polygon entirely inside another count as an intersection? And can the polygons have holes?

Regardless, a natural lower bound for running times for the problem is $\Omega(n^{4/3})$ where $n$ is the total number of segments in the input. Any faster and we could also improve the known time bounds for Hopcroft's problem of detecting an intersection between $n$ given points and $n$ given lines: expand each point into a small polygon (not so large that it intersects any line it didn't before) and test whether any of these expanded points belongs to a nontrivial connected component. This lower bound is mentioned briefly in my paper "Testing Bipartiteness of Geometric Intersection Graphs" which uses it to argue that bipartiteness of intersection graphs is strictly easier than connectivity; it in turn cites Jeff Erickson's paper "New lower bounds for Hopcroft’s problem", which proves an $\Omega(n^{4/3})$ lower bound in a reasonable but limited model of computation.