There is no such thing as "Oxbridge material" both Oxford and Cambridge University are keen to dispel the preconceived ideas that this outdated phrase alludes to.
You can't possibly stereotype nearly 45,000 individuals (under and post graduates) and say they all have shared characteristics that other students who attend other universities don't.
Clearly there are certain prerequisites such as high academic achievement but all high achievers do not go to Oxbridge.
I wouldn't say that there is any single unique trait that a person must possess to attend.
If any student wants to apply and thinks they have a realistic chance of meeting the entry requirements for their chosen course then they should give it a go.
DD went to Cambridge and she is from a council estate in Merseyside, loves makeup, hair extensions and false nails and likes to watch reality tv such as Real Housewives and Say Yes To The Dress. She graduated last year with a law degree.