The word "school" has its roots in the Old English word "scola," which itself came from the late Latin word "schola." This word originally referred to a place of leisure, or a place for discussion and learning, not necessarily tied to a specific age group. Over time, the meaning shifted to encompass institutions of learning, and eventually the specific institutions we now associate with primary, secondary (high school), and higher education.
So, high school is called "school" because it's a type of school, inheriting the general meaning of "a place of learning" from its etymological ancestor. There wasn't a separate naming convention created; it simply fell under the umbrella term that had already existed for centuries.