Yes, several. Herodotus (living in the 5th century BCE) tells us about four common elements. Language, ancestry, cult and sacrifices, customs.
They understood each other's languages well enough to consider them variants or dialects of the same language; and whoever was outside of that dialectal continuum was a Barbaros. The various dialects are not that different from each other . 2. Into this ties the belief in a common ancestry from mythological heroes. As the world becomes more interconnected and mythology less important, this is an aspect that becomes less relevant.
Another element are shared gods and ways of sacrifices. There were regional differences and peculiarities, but they all agreed in some aspects and respected each other's gods and temples. For example, they accepted the authority of the Apolline oracle in Delphi; they recognized the Mysteries of Eleusis; they had a certain mythological corpus that they shared, like the stories about the Trojan war.
Then, there are shared customs like the religious/sportive events in Olympia, Delphi, Corinth and Nemea. Temples are similar in their design and the Greeks seem to enjoy the literature that is produced in the different cities.
Remember also that the cities outside of heartland Hellas are settlements or colonies from the cities inside of it; and some colonies had colonies themselves. So at least these offshoots should feel very much as being part of one big family with their mother and sister cities.
By the time of Hellenism, starting with the death of Alexander the Great, Greekness became associated with a certain education and way of live: using Greek language and literature, having an interest in Greek philosophy, living in a city.
All in all, I'd say language is the most important common element. The Barbarian is literally the one who doesn't speak proper Greek.