You might want to take a look at this article. Our modern notion of 'Western' is absolutely distinct from anything the Romans may have thought of themselves because it's colored by Enlightenment and industrial capitalist narrative. That article has a great thesis and a great deal to say, it deserves being read in full.

The Romans had their own distinction, which they inherited from the Greeks, like /Kartoffelplotz identified. One was Roman, a naturalized member of the empire, or not. In this way it was like our modern concept of 'Western' in that it had more to do with cultural distinctions, and belonging to a narrative. That said, the Romans did not have some coalition with other nations, this was an internal narrative which did not supersede the military and political boundaries of the Empire. The Empire acknowledged foreign powers which were also generally unaligned, fighting mostly for themselves.

However, with the rise of Christianity, you see a different split again - Christians and pagans, particularly Muslims, who define a similar East-West, multicultural/multiethnic distinction. This split is largely along Roman lines, unsurprisingly. This distinction is one-sided in a lot of ways, because you have the Egyptian Muslim sphere are opposed to the Mongol Muslim sphere, which overlaps with the Persian Muslim sphere. Then again, you have the internal Orthodox-Catholic schism, which undermines the distinction pretty significantly, followed by the further Orthodox-Catholic-Protestant schism, and you do not see a 'unified' Christian sphere again until probably the Treaty of Westphalia. At that point, however, it was Christans versus undifferentiated pagans moreso than Christians versus Muslism, and so it ceases to have really any political narrative, as colonial missionary activities are at the forefront. In the political narrative you have a fragmented imperial struggle, on and off the continent, which has no East-West distinction.

The rise and distinction of Imperial Russia, as it says in the article I linked you, really creates the modern Western narrative. Russia's insular Orthodox and authoritarian culture caused it to seriously diverge from virtually the whole rest of Europe. This is not to say Russia was somehow wrong or bad or backwards (that is a preconception of the West), simply that you have a trade of ideas amongst the Atlantic and central European nations which does not really interact with Russia. You can follow that thread through the USSR's history too.

They didn't see the world as east and west. They saw it as civilized and barbarians