You mean in the same way that Romans were worshipping Attila just because he was on their periphery and pretty much ruined the (western) roman empire?
The central aspect of this narrative is revitalisation, which is for obvious not attributed to Attila, who was mythologised as a purely destructive force. However, even the mythology of Attila does express some points of this narrative, because in his role as a "scourge of God", he reveals the decadence of the centre, and in his tearing down of the old structures appears to clear the way for later revitalising forces. A more typical version of this myth (and the constructive counter-point to Attila) would be Charles Martel and Charlemagne, who have been mythologised as a barbarian or at least quasi-barbarian who came down from the Germanic periphery to the salvation of the Mother Church and consequently the revitalising of "European civilisation".
It's a narrative which is generally less pronounced in European history, because Europeans have this weird obsession with linearity and finality, of history marching towards a cataclysmic end-point, while this sort of narrative tends to be most pronounced in cultures with a more cyclical view of history, like China. But, it still exists, as in the aforementioned Charlemagne mythology. We could even plausibly read the "Third Rome" myth in Russia as a depersonalised variation on the theme.
(Now let's all just sit back and wait for Dachs to come and give me a good empiricist bollocking.
)
The timeframe you are using is way too late for such a myth-based tradition to be maintained. Medieval arabs and persians would be rather more likely to know the difference between Alexander and one of the emperors of the Persian empire before the hellenistic era.
What "differences" are significant, though? The European myth of a "European" king conquering "Asia" is just one mythology among many, so there's no reason to assume that Asian chroniclers would reach a point at which they came to believe it. They understood "Europe" and "Asia" in geographical terms, not the civilisational ones developed by later Europeans, so there's no reason to assume that they would find any particular significance in it when writing about Alexander.
Unless you mean that they would have been aware Alexander was Greek rather than Persian, which is true enough, but they were also aware that Cyrus was Persian rather than Babylonian or Egyptian, that Muhammad was Arab rather than Persian or Greek, etc, etc., and that didn't lend any civilisational dimension., That presumes to begin with that Greeks are fundamentally distinct from Persian in a way which Persians are not from Arabs, Egyptians, Babylonians, etc., which presumes a concept of "European civilisation" as set against "Asian civilisation" that, as noted, Asian chroniclers were not likely to have.