The origin of both 'Europa' and 'Asia' are largely attributable to their use by the Greeks. In the minds of the Greeks, Asia represented at first just the lands on the other side of the Aegean, while through their contact with the Persian Empire it slowly expanded to include areas of Anatolia, and then eventually, through usage, both it and Europe continued to expand (as did, Africa which only used to refer to North Africa west of Cyrenica).

That said, we have to be careful not to apply anachronistic ethnic thinking when we approach the idea of the identity. People weren't really identified with ethnicities so much as culture, and for the Romans and Greeks, anyone outside of their culture was an other. There were distinctions between Celts and Egyptians of course, but for the Romans, anyone who wasn't a 'Roman' was to be looked down upon. It is the same for the Greeks. The association with a geographic area is secondary; a Roman in Greece was still Roman, while a Greek in Asia is still Greek. We also have to add on to this that 'Roman' and 'Greek' are not really interchangable ideas, since one refers to an identity based on a polity, while another refers to a larger linguistic/cultural based identity. More accurate would be 'Roman' and 'Athenian' or 'Spartan', or 'Latin' and 'Greek'. For both the Latins and the Greeks, the city/polis was their primary method of identification. You were a Roman first, while all other Latins were of course related, but beneath you. Or an Athenian first, while all other Greeks were related, but beneath you. Or even an Arician (Latin League member) first.

If you consider this, the Romans had as much in common in their mind with the Germans as they did with the Egyptians or Phoenicians. In this case, the fact that one was closer, and even perhaps closer in skin colour (debatable) is entirely irrelevant.

As for the idea of European identity, this is a more complex question. We can begin to see a European identity emerge in conjunction with the rise of a 'Christian' identity, which is to say, it emerges in opposition or in contrast to the perceived 'Muslim' identity and, and even more specifically, a 'Catholic' identity as compared to the Orthodox one as time goes on. The Cordoba Caliphate was in Europe, but would not be associated with identity. Yet, many of the people who lived in that polity have descendants who live in modern Spain who would, despite the 'moorish' blood that they retain. Then we have the development of racial ideology in the late Enlightenment era, and a more concrete idea of 'European' identity through writers like Rousseau, leading to the development of ideas like 'European exceptionalism'. But even here, Eastern Europe, and specifically Russia, is often left out, hinting at the origin of this cultural idea with Catholicism.

This answer is good but lack to show that at some points, the concept of europe as opposed to asia played a real role to identifiy the ennemy. It was first used to represent the Achaemenid empire known to the greeks as "king of asia". It associated a lot of stereoptype on the asiatics, unmanliness, laziness, they are born to slavery.... Those stereotypes, which fitted for the persian wars were reused in the following eras: by the Ptolemies and then the romans against the Seleucids and then the Parthian (two kingdoms wich existed on the land of the Achaemenid empire, and who had reused at least part of Achaemenid royal practice). This continental division was used in geography or for those stereotypes. It is also useful in the mix of the two, because the germans have different barbaric traits than the persians.

I'd like to agree with u/liwios and expand upon his answer. There was definitely a cultural divide between Asia and Europe, East and West, and it can largely be traced back to the Greco-Persian Wars and their primary historian, Herodotus.

The Greeks developed an identity based around their perceived opposition to the many Asiatic empires active in the Near East prior to the rise of Greece: the Assyrians, the Babylonians, the Medes, the Phoenicians, and finally the Persians (Achaemenid). As previously stated, the Greeks viewed these "Eastern" peoples as servile, decadent, feminine, while viewing themselves as independent, masculine, martial and simple (i.e. as simple farmers and rustic peoples, unaccustomed to the lavish wealth and decadence of the more fertile lands of Asia Minor--which had been under the sway of one empire or another pretty much since the Sumerians; and thus their populations were relatively more cowed than the less-conquered Achaean). The Greeks even developed a pseudo-scientific rationale for the differences they saw: the hot and flat open expanses of the Near East bred soft, effeminate people; while the rough and mountainous Grecian Europe produced the opposite.

And while the Greeks still viewed the Germanic and Celtic peoples as "barbarians" (those who spoke babble, "bar-bar" as it sounded to their ears) and certainly culturally inferior to the Greeks, they still saw in them familiar traits, such as "manly" martial prowess, simple lifestyles among harsh and rugged environments and within small local populaces, who do not try to impose their own cultures and lifestyles upon others or attempt to form vast empires (though they may very well attack your city-state and do some looting and slave-taking).

However, this "East vs West" mentality was not universal in Greece; far from it. Herodotus himself greatly admired Asiatic peoples, heaping much praise upon the Persians in his "Histories". Even Thucydides viewed the differences between "Hellenes" and others as minor and relatively inconsequential.

Eventually, the Athenian Empire would come to be seen to resemble some "Asiatic" qualities to many Greeks, until the Peloponnesian Wars destroyed both its and Sparta's power, leaving the region open for conquest by Macedonia, and the creation of the first true "Eastern" universal empire in the West by Alexander--largely the inspiration for the later Romans in creating their own empire.