Sorry it took so long for me to get to this.

The pivotal work on this topic is by George Makdisi, The Rise of Colleges: Institutions of Learning in Islam and the West. It's a bit old, and it's been critiqued heavily, but it's still a good starting point (say I).

The simple answer to your question is this: all education was religious. There was one kind of learning institution--a madrasa--and all students learned religious law. From there, they could specialize in various fields; the idea was that doing research into science, math, medicine, astronomy, etc. was discovering the intricacies and wonder of the universe that God made, so by doing this kind of research you were placing the universe into a religious context.

As Makdisi describes it, there were eventually two nebulous fields of specialization: 'ilm, which is what we would call the hard sciences (biology, botany, medicine, astronomy, etc.), and adab, or the humanities: philosophy, literature, history, etc.

However, it's important not to think of these as majors in the way that we understand them today (or, in the UK, the idea that you would chose one of these subjects to read in) -- students at the madrasa would have been exposed to all of these and might not have seen them as entirely distinct areas of study.

As part of their further study after receiving their degree, they would have engaged on research and experimentation with a scholar in the particular area they wanted to work with.

That's why you get these people who seem to have an endless list of specializations behind them: "he was a philosopher, mathematician, spoke 27 languages, and modeled the orbit of Venus in his spare time". Everyone was, to some degree, a philosopher; the orbit of Venus is calculated mathematically, and he probably learned to read all those languages so that he could read the mathematical research on calculating orbits (and the philosophy behind heavenly bodies) in their original languages.