What made the ancient Greeks Greek?

Answer 1

They were a collection of similar peoples who spoke a common language, shared a pantheon of gods and a mythical and legendary tradition. They formed a religious and cultural (not political) group - the Amphyctionic Council which they limited to what they believed were Greek tribes.

Others were 'barbarians' ie not Greek who spoke different languages. The Macedonians were not recognised as Greek, and Alexander the great's father Phillip II had to fake an ancestry linked to a Greek hero to allow him to enter a chariot in the Olympic Games. And in order to gain representation on the Ampytionic Council he had to bribe small member cities to be his proxies so that he could influence it.

Greek is of course not their word - they called themselves Hellenes after a traditional founder Hellen. They today call themselves Hellenes. The word Greek came from a tribe in Italy - the Graeci. As the early Romans pushed down the Italian peninsula, they ran into the Graeci. As they went further south they ran into the Hellenic cities around the Italian southern coast, and thinking them to be a lot more Graeci, called them the Great Graeci. The name Greek has stuck for all Hellenes.

Answer 2

The Greeks recognised as Greek anyone was a citizen of a Greek city-state.