China’s chaotic era of disunity is referred as such, why not Europe?
It is often difficult to offer (somewhat) definitive answer to "Why not....." type question, and I'm afraid this might be one of such sort.
First of all,"warring states" period [as a periodization] is a rather literal translation of the Chinese original term, 戰國時代, to denote the concrete period. As /u/ParallelPain recently illustrated in: Would I be right in assuming that the Japanese Warring States period's name (Sengoku Jidai 戦国時代) is an intentional allusion to the much earlier Chinese period of the same name (Zhànguó Shídài 戰國時代)? When, how, and by who was this period named?, "Sengoku Jidai (Period)" (戦国時代) in Japan also borrow its name from this Chinese periodization by the 16th century Japanese, and this borrowing presupposed the learning of Chinese classics like Stratagems of the Warring States (Zhan Guo Ce : 『戰國策』).
When the periodization of the European Middle Ages took shape in course of long early modern period (by about 1700), these classics as well as the knowledge of ancient Chinese history had not been acknowledged widely among European intellectuals yet - in fact, Martino Martini's Sinicæ Historiæ Decas Prima (1658), the first European work introduces ancient Chinese history (from legendary three sovereigns and Five emperors down to the birth of Jesus as a history of human) divided the opinions of European scholars in the late 17th century. While Leibniz was rather positive on the alleged ancient chronology of Chinese history that apparently predated Noah's Flood in OT, some scholars like Paschal (in his Pensées) was very skeptical of the authenticity of this claim (Okazaki 1996: 146-57).
It is true that the conflict between the universal history based on the Bible and legendary history was the main issue of this dispute, but the authenticity of ancient Chinese history was certainly still disputed when the famous three periodization (ancient/ medieval/ modern) was being formulated in European university. Thus, it would be very natural not to borrow the term from ancient Chinese history instead of traditional European one, I suppose.
There has also been another, conceptual problem to adapt this term (in theory) either in history [especially of medieval and early modern political thought] or in international relations - that is to say, "Was there state(hood) [state as a political concept] in (especially early) medieval Europe?"
While being often disputed even in the beginning of the 21th century (to give an example, the debate between Davies vs Reynolds) on the medieval state, one of the classical but still popular grand narratives in European history is the long-term "formation" of the state (with enough machinery to mobilize the resource from the subject) from the Later Middle Ages to early modern period, and the possible role in which the war between the emerging states played, after the revival of the concepts of "public, politics" in the midst of the Middle Ages. In such a way of thought, the state was not the stable, essential conceptual component throughout the European Middle Ages, but being transformed into more modern / efficient one throughout the Middle Ages and beyond (Early Modern Period). While not directly named after "Warring States", the classic historical debate on "Military Revolution" in Early Modern Europe also focuses on the relationship between the war and the state formation, I suppose.
It is also worth remarking that neither the East Asian historians did translated the original (Western/ European Middle Ages as such [like the "warring state" period, as suggested by OP] in the 19th century. As I briefly mentioned before in: Is it proper to use the terms “medieval” or “middle ages” for areas outside of Europe? Are there more appropriate terms for this period in Asian and African history?, Japanese historians invented/ literally translated the Middle Ages as 中世 (lit. Middle Era) in the 19th century, but this new term of periodization also matched traditional East Asian vague concept of the past as well as the meaning of the original term closely (Kishimoto 1998: 20). They also understood that the primary meaning of the Middle Ages was a concept, rather than the concrete period.
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