I'm going to preface all of this by saying that every University was different, and that as time passed Universities tended to develop rather individualised traditions in different countries. If I have overlooked any special examples, I do apologize.

Generally, no. Scholasticism, the governing philosophy of early Universities, taught the Seven Liberal Arts: Grammar, Rhetoric, Logic, Arithmetic, Geometry, Music, and Astronomy. The first three were called the Trivium and made up the curriculum of a Bachelors Degree. The Trivium was seen as preparation for studying the more difficult Quadrivium (the last four subjects). Once a student had completed the Quadrivium they were considered ready to teach at a University level, and granted a Masters Degree. Unlike modern Universities, there weren't exactly majors or minors, and in the early stages of the University there weren't individualized departments. "Faculties" of Law, Theology, Medicine, and other subjects gradually materialized.

So instead of choosing a major, fulfilling its requirements, and then taking interesting electives on the side like we do now, medieval students had to first choose a University that specialized in whatever subject they were interested in. For example, Paris was well-known for theology, Bologna was well-known for Law, and Salerno was well-known for Medicine. At University, students sometimes chose their own professors and paid them directly, or sometimes they were assigned professors and paid tuition to a central authority. The caveat was that they could only really study the Liberal Arts.

That being said, medieval intellectuals were much less confined by disciplinary boundaries once they took their Masters than modern scholars are. A scholar with enough experience and respect could research, think, and write about almost whatever they wanted.

NOTES: Paul Oldfield "The Kingdom of Sicily and the Early University Movement" Peter Denley "Medieval Italian Universities and the Role of Foreign Scholarship" A.B. Coban "Decentralized Teaching in the Medieval English Universities"