Most questions about cluster analysis seem to come from people who have a single dataset and want to compare/quantify the similarity of different clustering approaches. This question is not that. Instead, my goal is to take two separate datasets, apply the same clustering technique, and then compare/quantify the similarity of the resulting clusters.
I'll make it a bit more concrete. Let's say I'm taking a survey, and I want to use hierarchical cluster analysis to reveal the latent groupings in the data. Survey A takes an hour to administer, whereas Survey B takes 5 minutes to administer. I can probably assume that the data obtained from Survey A are a better estimate of the real world, but I want to know how well Survey B stacks up. Clearly the actual numbers are going to differ, but if both surveys yield the same clusters, then it's probably better to just use the shorter one.
So the big question is: what's an appropriate metric for measuring how different two sets of clusters are? I've had a quick read through Comparing Clusters - An Overview, and my first sub-question is whether there's been subsequent development since this paper was written (2007). They tentatively advocate for measures based on mutual information (section 5), but then caution that it's not well worked-out. My second sub-question is whether it's even appropriate to apply these methods to clusters that are based on different datasets.