Some have touched on school in the ancient world so, I'll go to the Medieval West, starting in around the 8th/9th century and beyond for a bit...

School was most often run by the church or church adjacent institutions [like a monastery or, a group of monks/priests in the employ of a King, say] and as such would require study or religious texts, as well as the "seven liberal arts" - the Trivium and the Quadrivium, if you've heard of those.

The trivium was grammar, logic, and rhetoric. In studying this, most fledgling monks or lay people would probably be reading a lot of Aristotle and perhaps a smattering of Cicero or other Latin writers. Meanwhile, the quadrivium included arithmetic, astronomy, geometry, and music. Here again, classical authors would have been read and preserved, along with some late antique and medieval writers as well.

Starting in around the 11th/12th centuries, monastic schools started to develop into larger institutions, which eventually gave rise to the earliest universities [oversimplifying some]. The seven liberal arts still made up the core of one's education which could lead to a bachelor's degree. A Master's degree might focus more on the quadrivium, or an area within that, and after attaining an MA, one could study theology, medicine, or law at higher levels.

So as you can sort of see - you wouldn't see people studying history as its own field. History would come up when studying theology, when reading for rhetoric and grammar, and in other ways tangentially, but it was not a focus.