The Greeks had an immense sense of pride as a people, but the idea of a single country isn't really something we can really apply to their world. I'll go through a few categories of similarity etc. here, but it's crucial to remember that these categories were not distinct in the same way that they are in our world.

The Greek people can be split pretty safely into three ethnic groups, the Dorians, the Ionians, the Achaeans and the Aeolians. The Dorian city-states tend to be in the Peloponnese (like Sparta) and claim some descent from Heracles. Ionians, like the Athenians lived around the Aegean, from Greece to modern day Turkey and Syria. Cities that identified as Aeolian tended to be on the western side of Northern Greece. Sometimes their linguistic and cultural differences contributed to conflict, like in the Peloponnesian war, but histories have been written that don't refer to these differences once (including those of ancient times). My point in all this is show how Greek identity had many levels, a Theban could at once be Aeolian and different, or a Greek and part of a wider whole.

This is even more evident in the world of religion and myth. Zeus was the king of the gods, and all Greeks worshipped him, but Zeus is observed in many forms, and individual places had special affinities with the god. The Elians presided over the sanctuary at Olympus, and had idiosyncratic rituals, but it was also a panhellenic divine place that the city-states gathered at to celebrate their Greekness with games.

You seem interested in warfare and alliances, it's the crux of your question really, but it's important to remember that war fits into a shared cultural heritage too. There isn't really a Greek word for peace in the way we use it, they seem to refer to the status quo as stasis, which has a sense of constant movement in it, and ironically can mean both normal affairs and a conflict. They saw their (pre-Peloponnesian war at least) combat as an unavoidable part of reality. We can say that this was for many socio-cultural reasons, but in their mind it must have been very different.

Yet, as you point out, there were many leagues and alliances that often verged on empires. The Athenian Empire/ Delian League that emerged after the Persian Wars is definitely the example you're looking for, there were all sorts of economic, religious, social and political changes forced upon the island states of the Aegean by Athens in this time, but I've put the other points in first to stress the different perspectives the Greek people had on such ideas.