This answer takes a deliberately pessimistic, devil's advocate tack. I don't actually think we're all that cliquish, I just want to challenge a few assumptions here and there.
Taking your questions out of order:
How can a Q&A site have the social dynamics necessary for the formation of cliques?
Any time you get enough humans together and ask them to accomplish something, cliques will form. It is inevitable. The question is not how you avoid cliques. The question is how you can leverage them to make the group as a whole more welcoming instead of less welcoming.
What is currently going on that would make our site seem exclusive?
Having said all that, cliques can form in a number of different ways. On the one hand, you can have natural divisions arising from meaningful and important differences, and on the other hand, you can have artificial divisions imposed by social norms, based on whatever arbitrary factor happens to be convenient (race, gender, etc.). Divisions may also be horizontal (tags, tag clusters) or vertical (rep, privileges).
It's my opinion that natural divisions are less problematic than artificial divisions, and horizontal divisions are less problematic than vertical divisions. The existence of rep is a vertical division, and somewhat arbitrary (Why is an upvote worth 10 points? Because that's just How It Works), but rep is not meaningless because it is earned rather than given. Nevertheless, the stratification may prove intimidating to new users, or suggest to them that the clique of high-rep users is in some sense "better" or more important than everyone else. But on the other hand, this is just as true of any SE site.
Part of the difference between us and, say, Worldbuilding.SE, is that many of our questions have definitive right answers. Moreover, many of these questions are frankly trivia ("Did character X ever do thing Y?" and the like). This can make the rep divide a bit more intimidating than it might otherwise be because:
- It exacerbates the FGITW problem (a disproportionate share of votes go to the first answer rather than the best answer).
- It advertises the encyclopedic level of knowledge of many of our top-rep users.
- Both these things contribute to the impression that new users cannot compete with experienced users either in time or in comprehensiveness.
Given that Math.SE handles such "hard questions" on a regular basis, they might be a more interesting contrast than Worldbuilding:
- They are incredibly friendly towards less mathematically experienced users. They will happily write a lengthy answer to explain a person's simple misconception in great detail.
- They often try to answer simple questions with hints rather than with definite answers, to encourage the asker to figure it out on their own.
- They have the soft-question tag, which allows for much more subjective questions than you see on most SE sites, especially relative to the strict black-and-white nature of mathematics itself.
If SFF.se is "clique-y", what can individual users do to alleviate this impression? Do we even want to try?
One solution that I have seen work elsewhere is to welcome new people to a particular clique (tag cluster) in addition to welcoming them to the group (site) as a whole. This welcome should be particular to each clique, and provide resources which are uniquely helpful to new members of that clique (for example, you might welcome a Harry Potter fan with links to Mugglenet, Accio Quote, Pottermore, etc.). I'm not certain how well this would work on a SE site, but it's something to consider.
Regardless, experienced users should try to imagine the site from the perspective of someone who's never even heard of Stack Exchange, or whose only exposure to it is Googling things and reading the answers. There are quite a lot of rules (e.g. don't post duplicates) and social norms (e.g. downvotes aren't personal) which are not at all obvious to someone who hasn't spent a significant amount of time lurking at the site. Try to be patient with new people who have not learned these things, but accept that you will not reach everyone.