“You are not in Chinese High School, you are in THE Chinese High School.”
Indeed, it is a quirk by the principal, but what’s the intention of adding “the” to a noun?
The definite article is used before singular and plural nouns when the noun is specific or particular. The signals that the noun is definite, that it refers to a particular member of a group.
So in this case, the principal was trying to stir up school spirit by telling his students they are studying in the one and only Chinese High.
In everyday use, “the” is important when you are referring to a specific object/incident.
A dog bit me. (that can mean any random dog.)
The dog bit me. (you know which dog it was and you are referring to it.)