Westminster College was a London-based college, before relocating to Oxford. For a short number of years the Bachelor of Education from Westminster College (and, I think, the Bachelor of Theology) was awarded by Oxford University. Westminster students were allowed to become full members of the Oxford Union (unlike Oxford Polytechnic students). They were members of the Bodleian Library. They could join Oxford University clubs and attended Oxford University lectures. As you say, Westminster scarves and ties were sold alongside other colleges. They were non-matriculated Oxford University students.
Had a "financial deal" not been struck it is widely believed that Wesminster would have gradually moved towards becoming a full college of the University. So, those who studies before or after the link with Oxford University are not Oxford Alumni, whilst those who did enjoy that link are.
Not all that strange, really.