I was told in my European history class that the idea of a "European" is a relatively modern concept, and that for centuries before, people in Europe post roman empire were more likely to consider themselves part of "Christendom" rather than part of Europe. This made me wonder, would someone from let's say Western or Central Europe consider the Byzantines and other Orthodox groups part of Christendom, or as something separate?
This is a bit complicated because it gets into questions of historical identity, which are usually fraught and debated. “Europe” as a term was originally just a word that the Greeks used for the areas they inhabited in the Balkans and was only later applied to the rest of what we now call Europe. The degree to which anyone living in Europe thought of themselves as “European” is hard to figure out, but your teacher is probably right that that wasn’t considered a defining characteristic for individuals. Physical identity was probably more based around your ‘ethnos’ (Greek word for ‘people’ or ‘race’; I use it here because it doesn’t have the modern connotations of race), such as ‘Frank,’ or ‘Saxon,’ or your city of origin. There isn’t much evidence that people felt they belonged to a greater geographical continuum, but again, this is a heavily debated topic, so take that with a grain of salt.
As far as the idea of Christendom as a uniting factor, this too is contentious as a unifying factor, but it does seem like people considered themselves as part of a greater Christian whole, at least in our written (elite) sources. Byzantium was absolutely considered a part of Christendom, at least until 1054 when the Eastern and Western churches were in communion and at least technically unified in belief (not going to get into filioque or azymes here). After the so-called ‘Great Schism,’ there is a cooling of relations, for sure, but not really the rhetoric of the Byzantines as ‘non-Christians.’ We see in the early Crusader texts that there is a lot of hope that the ‘fellow Christians’ in Byzantium will help out the Crusaders, and even when they don’t provide the help that the Crusaders want, there isn’t much rhetoric of ‘non-Christianity.’ Much later, when the Ottomans conquer Constantinople, there is a lot of concern in Europe for the destruction of a Christian Empire in the East and what that means for the rest of Christendom. They may be bad Christians, even heretical Christians, but they are still a part of Christendom as a whole. This is not to say that there are not divisions (there is much talk about the ‘Latin Christians’ vs. the ‘Greek Christians), but I think the main thrust of your question was just about their general Christianness.