It can't be from the City of Oxford, as the arms are wrong.
Webster's College Dictionary says:
tip•staff (ˈtɪpˌstæf, -ˌstɑf)
n., pl. -staves (-ˌsteɪvz)
-staffs.
1. an attendant or crier in a court of law.
2. a staff tipped with metal, formerly carried as a badge of office, as by a constable.
3. any official who carried such a staff.
Other dictionaries say much the same.
That was why I suggested the Oxford City Police?
The addition of the anchors makes me think it might have been the River Authority, or River Police?
The coronets (NB not crowns!) look to me like civic coronets? But as they are a charge on the field, rather than a crest, they could mean almost anything?!

I tried all the colleges at Oxford University, but none of their Arms are even close.
In summary, I would say it's a civic office, connected to Oxford City.