Does it not depend upon whether you are doing an Arts or a Sciences subject? As an 'Arts person', I meet my supervisor fairly regularly, but my work remains my own, i.e. I bounce ideas off them, but try not to be influenced too much by their methodologies. It's fairly informal. I can imagine, though, if you are a scientist, that your supervisor's input is much more influential.

Re: the Oxbridge thing. I was offered the opportunity to remain at Oxford to study as a postgrad. I turned it down purely because I couldn't afford their fees and am now at a redbrick. I work very well with my supervisor, but I still wish that I could have remained at Oxford. Better networking and career prospects, methinks.