When you look at Greek history since antiquity, they seemed to be far more "Eastern" than Western. When you look at what part of the world they were in contact with, and involved in the most, it has always been the East, The city states were in contact with the Persians, and involved in conflicts with them, Alexander the great spread Hellenism through the middle east into Asia, Not Germany or Norway, Or great Britain, Greeks set up their centers of leaning in places like Alexandria, in Egypt, not in France, or Belgium, Ancient Greek culture seemed to have far more in common with Persian, Egyptian, Sumerian, Phoenicians etc, than they did with any Northern or Western European tribes
During the middle ages, the Eastern Roman Empire, was the Greek continued roman empire in the east, and it was involved primarily in the east, and influenced mostly the eastern world. Not just in the middle east, but also in Eastern Europe, in places like Russia, The people who arguably preserved Hellenism, were the Byzantines and then later the Muslims, not the Western World.
So I don't understand why Greece is "Western History" I can see how Hellenism inspired Western Culture, but just because something inspires something doesn't make it part of it. I would say that countries like Russia were far more influenced by Hellenism and Hellenic culture, than say France or Britain, especially during the Byzantine period, and their influence on Rus
It seems a bit of an appropriation by Westerners honestly, even in the way Ancient Greece is depicted in Art, and even Movies, Extremely pale, fair haired Anglo people sitting around in Ancient Garb, discussing philosophy in British accents.
So why Is Greek history, and Greek culture, always referred to as being "Western" Despite Greeks being vastly more involved in the Eastern World.
The answer to this question should not be sought in Ancient Greek or even Roman history. Rather, you should look to renaissance and early-modern or 19th century Europe and the European historians who first came up with these ideas.
History is constructed. Events happened, facts exist, but the way we frame and interpret and conceptualise them is more a product of the historian than of the past he describes. Greece is referred to as being "Western" because the story of Western civilisation is held to begin with the Greeks. But the people making this claim are, on some level, not inherently interested in the story of Greece. They're interested in 'the' story of Germany, or France, or England, or America, and they want to know what the story of Greece has to say about those stories.
Greek civilisation and culture did, of course, influence western Europe, albeit mostly indirectly. You could argue (and I would agree) that this influence is overstated. But in a certain way, that doesn't really matter, because the idea that western civilisation started with the Greeks has been an incredibly powerful one for centuries, and regardless of whether it's true or not, that idea has had a tremendous influence. See for example Victor Hanson's work ("Western way of war" et. al.) if you want to know how that idea is still used by some modern western thinkers.
We recently had a very interesting discussion on this topic, or rather its inverted form, by people who know more than I do: why doesn't the Middle East want to claim a similar connection with Greek history? I highly recommend you read it.
Greece itself as a Mediterranean power was superceded by Rome, who eventually conquered Carthage to become the sole power of "Mare Nostrum". Much of Rome's culture including its religious pantheon and ethical and philosophical groundings were rooted in the Greeks. This is why when historians refer to the origins of Western civilization they will often use the term "Greco-Roman civilization". Greek civilization played such a prominent role in shaping Roman ethics and culture that the two became somewhat syncretic.
While Rome certainly did not adopt democractic governance in the republic anywhere near the scale of athens, it did play an influence. Attic Democracy was most influential in the renaissance, where there was something of a Hellenic revival with the discovery of texts brought via the fleeing Byzantine scholars & The Ottomans.
It's worth mentioning that Hellenic culture was not alien to the Italian Peninsula or elsewhere in Europe before being adopted and spread by Rome. The Greeks themselves had their own colonies throughout Europe, stretching from Crimea to Iberia and trading with the local tribes for food. There is evidence that the Etruscans and those living in Latium were influenced by the Greeks as well, before Rome was founded.
For the Greek colonies, source is CM Bowra, Classical Greece.