Do Durham discriminate against Oxbridge applicants?

I think it is a fair question. If the UCAS forms do not show your other choices, then absent something you say in your personal statement, other than making your application early to meet Oxbridge deadlines, Durham cannot know and so cannot rationally discriminate (i.e. other than against their own interests in trying to identify the likely Oxbridge successful applicants!).
But I read law at Cambridge and had a rejection from Durham a few days after I received and accepted the offer (2 of my other back ups made offers too). This was 1990. I have found this quote on activelearning.ac.uk about whether other UCAS form universities get to know anything about where you have applied (as initially, no) : “However, universities do find out once you have accepted your offers or have no live offers available”.
I don’t know if that means that Durham can wait to see if other offers are accepted, and at that point only make offers to those without an Oxbridge offer (/accepted offer?). THAT might be a rational strategy to minimise uncertainty in their offers vs acceptances. Someone should be able to answer that question - it still wouldn’t mean do not apply to Durham (likely % you won’t get an Oxbridge offer), just that you are unlikely to get an Oxbridge offer and a Durham offer as a ‘back up’.

The last part wouldn’t work. Students can only accept offers once they’ve heard back from all their choices.