Can you take classes at both as a student at either?
Oxford and Cambridge are totally separate universities, 3 hours’ drive away from each other. You can’t cross-credit classes.
However, they do interact in a lot of other ways. The Oxford vs Cambridge varsity sports matches are the highlights of each team’s calendar - the Boat Race in particular draws a crowd of tens of thousands, while Rugby Varsity is broadcast live nationally. There’s also a lot of collaboration between professors and researchers at both institutions, as you would expect.