The Italians may not call themselves Romans, but most Europeans do attach some importance to the geographic area they are from as well as their linguistic/tribal identity, or whatever you want to call it.

For geography: On top of being European, the Eastern Europeans call themselves Eastern European, the Balkan Europeans call themselves Balkan, the Nordic North Europeans call themselves Nordic, and so on, after the larger geographic area that encompasses their country and other culturally similar. Kind of how the Lebanese, Syrians, Jordanians, and Palestinians are grouped together under Levantine.

For ethnic/linguistic identity: On top of being European and Eastern/Western/Southern/Northern European, most Europeans also identify as Slavic (in large parts of Eastern Europe), Nordic (in Northern Europe), Germanic (in parts of Western Europe), etc. A Polish person will have much more in common with a Lithuanian or a Czech than they will with an Italian or Spaniard with whom they don't share a culture. A Slav from Poland will have much more in common with a Slav from Serbia or Croatia than they will with someone from Italy or Spain. I imagine it's the same for people from the Levantine countries, who share at least some common culture vs. Levantines and people from, say, the Maghreb, who don't share as many cultural similarities.

But I get the Phoenician argument, the pan-Arab argument (in Europe, pan-Europeanism only works during Eurovision), and the Lebanese arguments. A Pole will see themselves as a Pole, a Slav, an Eastern European, and a European, and none of these identities are mutually exclusive. It's a bit different in the Middle East, though, because certain identities do clash.