A bit of confusion is resulting from the use of the words "black" and "white". Blacks in Britain are a minority, and many have strong cultural connections with particular parts of the world - Ghana, Nigeria, Jamaica for example. There are also certain shared cultural experiences that may be said to be "or black origin" e.g. the dominance of black artists in hip hop and RnB. As such, "black" is a metonym, which encompasses a cultural ecperience, not simply a race in the physical sense. Black thus refers more to a presumed cultural experience, which is not necessarily in toto comparable to the mainstream of society. Such groups provide a means for cultural interaction rather than segregation. So that's not to say white people don't listen to RnR or hip hop, or have no interest in matters relating to race, but the significant "discriminatory" criteria is cultural, not especially racialist. In summary, there is, in this country, a "black" cultural experience that is defined against a white majority background. This is so partly because minorities in a given country tend to lead their lives in dstinct social and demographic groups. Being a minority of any sort, is something that takes pace on a socially narrow front. In Ireland for example, many of the ascendancy British will share a similar story. Just as in the former colonies, the white minorities will have a shared experience that can't easily be separated from their status as minorities. They live similar lives, in a sense, or at least have some shared story that applied to them as a minority, and isn't especially relevant to the majority. It's partly a numbers thing.
I don't really know if "white" as a racial descriptor carries the same conotations of a shared (or a presumed) cultural experience. This is simply by virtue of the fact that, being a majority group, the cultural experience of being white and British is not as narrow as it would be for a minority group - there are few issues that arise in living in Britain that are exclusively or significantly the result of being white; whereas many that result from the very status of being gay, asia, black, female, a child, aged, a teacher or a christian. Furthermore, the opportunities to experience the majority culture, is so broad and ubiquitous that it is hard to justify on purely practical grounds, the need for a club which discriminated on the basis of race in the case of a "White Society". There would be no practical reason for that kind of implicit discrimination, as there is in minority clubs - of any sort, from trade unions to bridge-players societies. On what basis would they justify a discriminatory membership other than the fact that they did not want people of X race to be members? If there is no practical justification for a discriminatory policy it is often simply racist and/malicious. By way of analogyn, I don't see it as sexist for, say, a Woman Executives Society to exist, because the challenges that women face as a minority in many professions, are different from those that men face - and, most importantly, those challenges often arise as a direct or indirect result of their being women. In the same way, a Black Society would probably exist not to discriminate against others, but to try and share a particular cultural experience that is inextricably linked with being black in a largely white country. Let's be sensible about these things.