Is it possible to have the Roman and Greek empires to both dominate the western world in the classical age? Will there be some sort of a rivalry to possibly arise between the two ala Cold War?
What are the best PODs for this to happen?
Well the best POD would be Alexander the Great living up to the old age, dying in his bed when he is 64 years old. His army is already sworn to his son heir as the next basileus/shahanshah of the Great Empire. And the dynasty rules the Empire for a few centuries.
But that would be too fantastic.
The better looking POD is the Empire of Alexander the Great divided differently than in OTL. I mean in OTL the Seleucids were weak because of the insufficient powerbase - the Greek/Macedonian settlers colonists were too few, the non-Greek troops were unreliable and one bad defeat of the Grand Army was thus disastrous.
The Romans on the other hand could afford lose army after the army, totally annihilated, and then recruit new ones.
The Seleucids just did not have a chance in OTL. The same applies to Egypt if it stays "Greek" as in OTL and does not conscript native ethnic Egyptian troops.
So for the Greek/Macedonian Empire to oppose Rome it must have the power base - Macedon, Balkan Greece, (you might add Western Anatolia and Syria): the manpower resources to conscript new armies from Greek/Macedonian population if things went badly.
The best possible shape of this "Greek" Empire is the East Roman Empire of OTL (Balkans, Anatolia, Syria and Egypt). It would be nice to add Mesopotamia to make it perfect.
The minimum Greek Empire to have some chance to survive wars with Rome is Macedon, Balkan Greece plus Anatolia and Syria;
or less probable but still possible is Macedon, Balkan Greece plus Egypt.
But it must be Greek/Macedonian Empire which includes the above territories for a few generations in order to knit these parts together and have some kind of patriotism and loyalty.