Before I begin, please note that I am not troubled by the votes to close of my question here, but I'm somewhat puzzled about them. The purpose of this meta thread is for someone to explain why the question isn't about research-level mathematics.
To save some of your time:
1) I realize that MO should be reserved for technical questions related to specific spots in research where people are stuck, or need a reference.
2) The question could be "morphed" into a general question about any area: what are some intuition pumps for e.g. Zoology.
3) It may not be desirable to provide answers that "oversimplify" technical subjects, as a little knowledge is a dangerous thing.
My justification for posting the question despite the above points is as follows: (1) If MO is only about technical questions and not about more general strategies for conducting one's life as a research mathematician, it loses some appeal in the community. (2) Even with the morphability of the question, an expert's effort to intuitively organize an important idea in a subject can be useful for awareness of ideas outside one's own research area and can be regarded as meaningful research work (these intuitions are shared during talks, but if you don't work in an area, you may never attend such talks). (3) Regarding oversimplification, it is always a danger, but the purpose here is to promote awareness across fields and to promote unity. We may assume that practicing researchers won't think they really understand a technical area when hearing a high level intuition pump like this.
Perhaps the problem is that the sheer number of these intuition pumps is too large. I argue that the level of idea I'm looking for should be sufficiently high that the number of such ideas will be relatively small. The deformation/rigidity example is pivotal in my subject and has had enormous influence, so having this high-level analogy gives immediate feeling of a central movement in a subject. I'm looking for such broad things, to get a general feeling of the movement of mathematics as a whole. Maybe the reason to close this question is that such analogies can be provided in a "What is?" article in the Notices. This said, I haven't found such descriptions there.
I also must say that some of my friends don't really like the example I cite given by Popa. I am also puzzled about that. I can't see how such attempts to make things very memorable and salient are not helpful to mathematicians.
I did ask the question as an exploration of how mathematicians attach "intuition" to their techniques, and wonder to what degree experts work to find intuitions like these, and so won't be offended if the question is closed. This said, I'd like to record a solid argument for why such questions are not of interest to the mathematics research community.