Basically, my parents and their families are from different parts of Montenegro, I'm American, but they consider themselves Albanian (Illyrian) and that's what I've always grown up knowing and gone with whenever questioned. I've tried asking them to explain it before, but they've never really given a concrete answer; they just said that's what they are. I know this is very specific, but would like to hear if anyone has any insight. I don't know how many generations of their families have lived in Montenegro and, if it helps, they are catholic. Perhaps there is a historical or geographic reason for this, I don't know much history about the area or people, just that clans were big for a time and they love Skanderbeg. If this is the wrong place to ask this question, please tell me which subreddit I would have the best luck posting in.
Their ethnicity is Albanian which means they can speak Albanian and their ancestors were Albanians. Regardless nationality or religion, ethnic Albanians always identify themself as Albanians. In southern Montenegro and western Macedonia, there are cities with Albanian population majority.
Edit: Why do they live outside of Albania? Because of "London Conference of 1913"
As a result of the decisions taken and because of pressures from Greece and Serbia, half of the predominantly native Albanian territory of Independent Albania and between 30% and 40% of the Albanian population was left out of the newly established Principality of Albania. In particular Kosovo Vilayet was given to Serbia and Chameria to Greece.
Edit 2: You just have to read about Ulqin/Ulcinj for example, and you will understand everything better.
Albanian is an ethnicity. Montenegro is a multiethnic state comprised mainly of ethnic Montenegrins, Serbs, Bosniaks, and Albanians.