What role did religion play at universities in the Middle Ages?

At that time the Universities were Church establishments and students and master were classed as clerics (members of the Church hierarchy); they were "clerks in minor orders".

If you went to the University at Paris, for example, and peered in through a window you might at first think you were seeing monks sitting on the floor around a master sitting on his chair. In fact the students all had their heads shaved in the Roman tonsure, just like monks, but their gowns were different (cappae clausaeinstead of monastic habits) - these gowns have survived in a less medieval form in many Universities today.

The subjects taught at University were those approved and regulated by the Church, including many subjects with religious connections. It was intended, at least in the early days of universities, that the students would either become teachers themselves or would progress through the ranks of the Church to eventually become priests and bishops. Not all did, of course.