Something worth clarifying I think, is whether you wish to learn a European modern language, or an non-European modern language. Both have options in each area, however Oxford has a slightly wider range of subjects available through the degree these form part of (Oriental Studies) than Cambridge. At both you can combine a Middle Eastern language with an European language, although this is only noted in the degree name by Oxford; neither allows you to combine a modern Asian language with a European language.
There are also various other combinations, some available at both and some only at Oxford, if you wanted to combine a language with another subject.