There was absolutely a concept of pan-Greek identity.

Greeks never called themselves "Greek." This was a word which the Romans called Greek and has been adopted into western languages. Today Greece is officially called the Hellenic Republic. The ancient Greeks called themselves Hellenes. Amongst the Hellenes there was a further division into four distinct ethnic groups - Ionians, Dorians, Achaeans and Aeolians.

Examples of pan-Greek identity include: The Iliad, The Odyssey, the Panhellenic Games (a series of games held every four years which culminated in the Olympics. You had to be Greek to take part) and the Eleusinian Mysteries (for which you also had to be Greek).

Any shared Greek identity did not stop the Greek city states fighting numerous wars against each other. The Peloponnesian War was largely a war between the Ionian (Athens) and Dorian (Sparta) tribes.

The Greeks did fight together against the Persians. The Greco-Persian Wars (of which Marathon and Thermopylae are battles) were started because the Persians gained control of Ionia, a Greek region in modern Turkey. They instilled tyrants and the Ionians revolted with support from other Ionian peoples (Athens). Sparta didn't help the Ionians revolt because they themselves were Dorian.

After Alexander died, the resulting period become known as the Hellenistic Period. This period was marked by Hellenistic monarchy as Alexander's empire was split up into four different kingdoms. The city states of Athens and Sparta had no more political importance. Ptolemaic Egypt became a great sponsor of Hellenistic culture and many learned Greeks moved to Alexandria. I would say that the conquests of Phillip and Alexander strengthened pan-Hellenism because they spread Greek culture around the Mediterranean. They also diminished the power of the traditional city states who liked to fight among themselves. Since the Macedonians did not belong to any ethnic group, ethnic conflicts lessened.