I'm going to uni because I want to study my subject not because I want to make an impression on others. Although I suppose partly in a way its to try to impress future employers if you want to look at it like that but generally I would say no, not at all.
going to Oxford is often about networking because you can end up working with people later, and/or staying friends for life - but that's kind of true for anywhere. In the past when not many people went to Uni that was important. Sometimes though it's your parents who swan about telling people (seriously embarassing) that their son/daughter is at Oxford. In some circles like top independent schools it's what's expected rather than going to a little known new uni.
I think it is very pretentious for someone to actually lie about which uni they go to: essentially, it shows they think it overshadows the uni of whoever they're talking to. I would prefer a bragger over someone being quietly smug. But, realistically, very few are either.
However, of course, it all depends on the motive. I will be going to the "School of Oriental and African Studies, University of London", but inevitably, if asked in day-to-day life, I'll simply say I'm going to the University of London. It's too much fuss to explain it all.