But isn't there something fundamentally different about a group of people loosely associated around a particular thing they happen to have in common - not believing in god or liking a particular kind of sex or, for example, being a fan of a particular sports team or piece of media or whatever - than groups that lie on axes of oppression?

Good question! I find myself wondering about this distinction in regards to fat-phobia and 'thin privilege' too: is thinness / fatness it's own "axis of oppression," or is it just an issue lying at the intersection of ableism and sexism? And if the latter, does using rhetoric like "privilege" move the concept too far from its systemic origins?

a, atheists (by virtue of that identifier alone) don't have structural power (nor are they oppressed) and b, there is a genuine diversity of opinions and backgrounds within the community.

I think Atheism is actually a really complicated example, because (in the US) Christianity is certainly a structural reality (just like whiteness and straightness) that shapes US society: even life-long atheists tend to exhibit values that are at least in part the result of Christian understandings of the world (e.x. the split between mind and body). Likewise, just like non-whiteness, not expressing oneself as Christian can result in, in certain contexts, significant negative social consequences. There are plenty of Christian Privilege checklists! And while it is often said the 'the problem is the sexism/racism/homophobia/etc. in the Church, not Christianity itself,' that's by no means a decided issue. Check out Rosemary Radford Ruether in her seminal "Christology and Feminism: Can a Male Saviour Save Women ?": "This [sexism in Christianity] is no issue of passing and relative social forms. No emergence of women as equal to men in society can change the context of the discussion. For these [male theologian] writers, the exclusion of women from church leadership is not based on particular structures of society." Ruether is suggesting that the core of Christianity is sexist, and implies that feminism might ultimately need to leave Christianity completely behind. And the jury is still out on whether religion itself is oppressive--that is, maybe it's not just Christianity, maybe it's the very concept and structure of a faith system that's problematic (although I must mention that I think these arguments usually rely on an overly reductive definition of religion).

So is religion as deeply-rooted and naturalized as race? Comparing religious expression to race feels wrong, because, it's generally assumed, one can "pass" off one's atheist identity but one can't "pass" one's race... except that there are a number of examples of that happening!! e.x. Latinos (who identify as such) who pass as white (and conversely, Latinos who identify as "white" but white America identifies them visually as Latino!). Which is to say, just because atheists can pass doesn't necessarily mean that they aren't oppressed! Likewise, just as in the interesting racial tectonics in the Southwest US, diversity in the community isn't [necessarily] a compelling argument that oppression doesn't exist - I would never point to Latinos who pass as white and use them as proof that the racial constructs in the US are too fluid and diverse to be considered oppressive (to be clear: I know this isn't what you're saying, I'm just spelling it out to build my point).

So I think Atheists are at once (1) a meaningless grouping, because the only common characteristic is a lack of belief (2) a significant grouping as defined against the normative Christianity, indeed (maybe) an oppressed group (3) a cultural grouping defined by skepticism, maleness, and sexism.

I think someone could fit just one of these categories and reasonably identify as an Atheist, and I think someone who fits all three of these categories can reasonably identify as an Atheist. Also, my impromtu schema here is already showing its limitations: what about the "atheists and secular humanists who write at FreeThoughtBlogs"? I think you are right to suggest that that is a community big enough to warrant consideration as its own unique cultural brand of atheism. Perhaps there are others I don't know about!

TL;DR "Atheists" refers to several different and even distinct groups, which makes your questions about identity and community really hard to answer but also really fun to think about.