Yes the school code was the same for everyone at the University. It differs from college to college, since this is a "solution" that the university purchases from a company.
Not sure about the rest of the unis down south, but I definitely remember Cambridge and Oxford have some sort of "college" system which had no real relation to your subject (i.e. you could read Philosophy at Foo College, Oxford, or Bar College, Oxford). Maybe someone oxbridge-y can clarify.