No, it isn't. People outside the UK have this fascination with Eton. And don't seem to realise that there are hundreds of other independent schools, grammar schools (which are often as good or better than many independent schools), and (if we are talking about history) grant-maintained schools.
Eton isn't regarded as a particularly good school. It is relatively expensive, and preferred by people outside the UK. Their academic results aren't that strong.
And social connections are an aspect of almost every political party. Blair hired almost everyone in Derry Irvine's chambers. Most of the SNP are not only friends, they are actually related to each other. The Tories are possibly the least incestuous because Cameron made a concerted effort to widen the reach of the party in the mid-2000s (this is why the current Cabinet is so diverse).
The reason why there is a concentration of Oxbridge is because, when most of these people went to uni, they were far and away the best. In Scotland, you see the exact same thing with Edinburgh (and strangely, no-one talks about the fact that almost of Sturgeon's clique went to Glasgow...which is, generally, an inferior university). And what you are actually seeing now is that more and more politicians are going to other universities reflecting the fact that non-Oxbridge unis caught up in the 90/00s (and recently, they have dropped off far more...Imperial is ahead in sciences, Oxford PPE is still very good though).