I'm only familiar with my old college, King's in Cambridge. But patterns for Oxford and Cambridge are usually similar for most things.
When universities rent rooms it is usually their existing undergraduate student accommodation. This means it is available only outside of term time. Student accommodation is intended for single adults. There will mostly be one single bed ("twin" if you are North American). There will not be televisions or facilities like hotel rooms, although there will probably be a desk and a chair (since students have to work). There won't be "interconnected" rooms. There won't be room service of any kind. Breakfast will be canteen-style with the students. Be aware that you will be paying over the average price for the facilities you get.
Because of the style of rooms children will be difficult to accommodate. Even couples will find less accommodation than single people. Children are going to need to be at least old enough to sleep in their own self-contained apartment.
I did discover some exceptions by searching universityrooms.com (link above). The main one is Wycliffe College, which is a theological college in the north of Oxford. Its students are more likely to be married and have children than the typical undergraduate, and so it probably has a few family rooms for students.
You can do more searches to find alternatives. Colleges that work outside the usual pattern are more likely to have family style rooms.