Why are quatrains called quatrains?

The reason is the formal characteristics.

The quatrains are called quatrains because their formal characteristics are four sentences and one poem, short and pithy, concise in language, profound in artistic conception and rich in short chapters, which embodies the artistic characteristics of refined and implicit poetry. Jueju originated from the ballads of the Han, Wei, Southern and Northern Dynasties, and its name originated from the Southern Dynasties. Its origin can be traced back to Yuefu folk songs in Han Dynasty. There are strict metrical requirements for quatrains, which are divided into quatrains and archaic quatrains, among which quatrains have fixed metrical requirements.