I don't think there's any specific criterion which separates Classical from non-Classical societies or civilisations. It's largely a matter of convention and the context.
In conventional Western scholarship, "Classical Antiquity" refers to the great Mediterranean cultures from around 700BC to AD 500: particularly Greece and Rome. This is because of the enormous impact that these 2 cultures have had on our present world. Much of our current thinking about philosophy, logic, law, politics, language, literature, art, architecture etc still derives from Greek and Roman models (interestingly, the Western world has abandoned Greek and Roman music, after the rise of polyphony in the Middle Ages).
It's plainly a somewhat Eurocentric view. The Phoenecians were also a major Mediterranean society for much of the same period; they built colonies as far away as Spain and North Africa, and we have inherited the alphabet and many religious ideas from them (via their inland offshoot, the Israelites). But usually the Phoenicians don't get counted in the list of "Classical" civilisations.
This view probably comes from the influence of "Classicism". As Europe climbed out of the Middle Ages into the Renaissance and on into the Modern Period, models of Greece and Rome where often used as the basis for new ideas and a "return" to civilisation. Hence, "classical" is what was very esteemed. Indeed the word 'classical' itself comes from the Latin "classiscus" meaning "of the best kind, the highest quality". Roman architecture was considered "the best" - classical - and so on.
But "classical" is used in other contexts as well. We often speak of "Classical Chinese civilisation", meaning the Imperial period up to the Song Dynasty, before the Mongol conquest. There's nothing specific that separates the Classical Chinese period from post-classical; apart from a general sense that it was old, and in some ways a period of high achievement for Chinese civilisation.
Likewise, archaeologists and historians writing about American civilisations sometimes refer to the "classical Maya period" (c AD 250-900) to distinguish it from the early beginnings, or the period of decline from 900 until the Spanish Conquest.
So overall, I think some civilisations get called "Classical" because in the opinion of the person writing or speaking, they are "classic" :-) There's no other particular criterion, such as size, extend, duration, political organisation, etc.
Hope it helps!