In reply to:

surely 'oxbridge' itself must annoy one or the other, if not both...

This reminded me of a scene from "Blackadder Goes Forth" where Blackadder says that he deduced someone was a spy by listing hree great English universities - Oxford, Cambridge, and some redbrick which escapes my recall. He mentioned to his commander that of course only two of those were great universities, to which his CO replied "Too right. Cambridge is a bloody dump."