There are various systems of classical elements with five, four, three, two, or one elements: it would take an extraordinary ability with sources in many different ancient languages, though, to look at the evidence and see if there's any real kinship between them; or if, on the other hand, the non-Greek systems have only been identified as Systems (capital S) of classical elements because of resemblances to the Greek canon.

I'm aware that Vedic medicinal theory, for example, works on the basis of five 'elements', reportedly earth, fire, water, wind, and void. But is that a context-specific idea, or is it a taxonomy that's meant to apply to the cosmos as a whole? That's beyond my knowledge. Was this taxonomy seized on as a 'system' of classical elements because it genuinely is one, or because it's similar to the Greek system?

I have a sneaking suspicion that most of the world's five (or four, or three, etc.) element models have been seized on as 'systems' because of their resemblances to the Empedoklean model of four elements. As I said at the start, it's very hard to tell unless you're looking directly at primary sources in a wide variety of languages.

If the Greek system is the yardstick for the others, then it needs to be said that the Greek 'five element' system isn't something 'the Greeks' thought. It's an artefact of more than a millennium of afterthoughts and reinterpretations. The four-element system was devised by Empedokles in the 5th century BCE; the five-element system is a result of trying to synthesise that with Aristotle's thoughts on the nature of the sky. The idea of aether as an element, as such, is something that I don't see appearing prior to mediaeval English philosophy. It's based on something that genuinely is in Aristotle, but Aristotle isn't nearly as straightforward as he's usually made out to be.

Aristotle emphatically does not present a five-element canon. He does make it clear in On the sky book 1 (especially 270b) that the sky behaves differently from terrestrial materials. But On the sky is talking about classifying different types of natural motion, not about classifying substances. The idea is that everyday materials have a natural upward or downward motion, depending on whether they're light (fire, air) or heavy (water, earth), while the sky has a different natural motion, namely a circular motion.

When Aristotle does come to talking about classifying substances, in On coming-to-be and passing-away, he makes it very clear that he's actually firmly opposed to the Empedoklean canon of four elements. He explicitly calls it self-contradictory. In that treatise he weighs up various theories of primary substances -- the theories of the atomists, and the theories of people who thought in terms of primary elements like Empedokles -- and accepts some parts of their theories and rejects others. He accepts the Empedoklean canon only as a classification of properties that materials can possess -- wet, dry, hot, and cold -- while undifferentiated material is called hyle.

So I'd say it's a clear error to talk of Aristotle laying out a canon of four (or five) elements in On the sky, when he expressly rejects the theory in another.

Later interpreters of Aristotle had their own ideas, of course, and that's why I assigned the five-element canon to mediaeval English philosophers, who formalised the notion of a 'quintessence' or fifth element, identified with the aetherial region.

I say 'aetherial region' because that's how Aristotle talked about aether: as a region of the cosmos. Here's a report of a lost work of Aristotle (Cicero, On the nature of the gods 2.41-42):

Therefore, because the Fire of the sun is similar to those fires that exist in the bodies of animate creatures, it must be that the sun is also animate ... So since the origin of some creatures lies in Earth, others in Water, and others in Air, Aristotle thinks it absurd to imagine that no animal is generated in that element which is most suited to generating animate things.

A little later Cicero states that Aristotle talks about the aether as a region (aetherium locum). That is, Aristotle regarded the aether as a label for a part of the cosmos with a distinctive natural motion, and the celestial bodies in that region as fiery, not a distinct substance.

Mediaeval philosophers were split on the topic. Bartolomeus Anglicus (13th cent.) essentially repeats what Aristotle says about the sky behaving differently from terrestrial phenomena, and being unchanging (De rerum proprietatibus 8.5, trans. E. Grant):

[The aether is] something beyond the lunar globe that is of a separate nature from the nature of the inferior elements. Thus the aether is neither heavy nor light, neither rare nor dense, nor is it divisible by the penetration of another body. No corruption or alteration, universally or particularly, affects the aetherial nature, which would happen to it if its origin or composition were drawn from the elements.

Johannes de Sacrobosco's On the sphere (also 13th cent.) firmly identifies the sky as composed of a fifth element:

Around the elementary region revolves with continuous circular motion the ethereal, which is lucid and immune from all variation in its immutable essence. And it is called 'Fifth Essence' by the philosophers.

And Robert Grosseteste (12th-13th cent.) points out that if every kind of natural motion equated to a different substance, then there's a new problem, because the fixed stars and each of the seven planets have their own distinct motions (De generatione stellarum, introduction):

Every body is either homogenous or composite. A star is a body; therefore a star is homogenous or composite. But it is clear that it is not homogenous, because there are only five homogenous substances according to Aristotle and the other philosophers, that is the four elements and the noble substance, which is called 'quintessence'. But a star is not made of quintessence; nor is a star a single element, since if it were, the seven planets would constitute seven elements according to their nature, and that is not the case.

The upshot is that the idea of 'the Greeks' having a canon of five elements is really a systematising imposition on Aristotle. What he really thought was that the sky behaves differently from terrestrial substances; that terrestrial substances are usually classified as four elements, but they aren't really; and that stars are fiery. Systematising that into a taxonomy of five elements is an artefact of later thought.

Grosseteste's reference to 'the other philosophers' suggests that the idea of quintessence goes back further, but it isn't easy to trace. It isn't in Near Eastern interpretations of Aristotle, as far as I can find: Averroes' commentaries on On the sky don't seem to include a five-element taxonomy as far as I can find, and Grosseteste himself cites a Persian astronomer, Albumazar, for the different behaviours of celestial bodies. For the English philosophers I've relied on Edward Grant's Planets, stars, and orbs: the mediaeval cosmos, 1200-1687 (Cambridge, 1996), especially pages 422-428.