Why did Albania become so "different" from the rest of Europe?

Let’s differentiate some things: Albania as a country is not the same as Albanians as a people or Albanian as a language. There are overlaps, but just like with every country, not everything fits within neat borders.

First of all, Albania as a country is relatively recent, only declared independence from the Ottoman Empire in 1912. The borders of the new country were initially not accepted in the Conference of London of 1912-1913, so they were redrawn leaving a significant number of Albanian people outside the national borders, most notably in the region today known as Kosovo, which is why a majority of people from Kosovo identify as Albanian too. Kosovo as a country is even more recent, they declared independence from Serbia in 2008, a move that’s controversial to say the least. Both of them have differences because they lived under different systems for over a century, but they are still considered the same ethnic group. Yet, the cultural differences are there, so you have to be more specific on what do you mean. Something that you percieve as distinctly Albanian, might be a Kosovo Albanian feature, not Albania proper.

Secondly, the Albanian people have been first mentioned under this name since the 11th cenutry AD in Bulgarian sources as “half-believers”, i.e non-Orthodox Christians and we are told that its speakers already made a distict community from their neighbours Croatians, Bulgarians and Greeks. No information given on the other neighbours, Serbs, someone with more knowledge can expand on this:

can be seen that there are various languages on earth. Of them, there are five Orthodox languages: Bulgarian, Greek, Syrian, Iberian (Georgian) and Russian. Three of these have Orthodox alphabets: Greek, Bulgarian and Iberian. There are twelve languages of half-believers: Alamanians, Franks, Magyars (Hungarians), Indians, Jacobites, Armenians, Saxons, Lechs (Poles), Arbanasi (Albanians), Croatians, Hizi, Germans.

while in Byzantine sources as former allies of Byzantines. Albanians were described as living in what we today can approximate it as Eastern Adriatic coast, which today includes Central and Northern Albania, Montenegro, Bosnia, Croatia and Slovenia:

Unfortunately, the people who had once been our allies and who possessed the same rights as citizens and the same religion, i.e. the Albanians and the Latins, who live in the Italian regions of our Empire beyond Western Rome, quite suddenly became enemies when Michael Dokenianos insanely directed his command against their leaders...

We aren’t given any more information on where exactly in the Eastern Adriatic though, but there is a community of Arbanasi people as north as Zadar region of Croatia (Wikipedia Link if r/AskHistorians mods will allow, just to show their existence), while Albanian epic verse treats Udbina (in Croatia), as the northernmost “border” where Albanians could be found. While this is not a definitive answer to anything, it shows Albanians have historically have been rather spread out, more reason we cannot limit the history of Albanians only to the country of Albania.

Thirdly, there is also Albanian as a language. Despite being a language that was written down relatively late, there has been a lot of reasearch on Albanian language.

For example, it is generally accepted that Albanian language was spoken in the Balkans before Slavic migrations in the 6th century AD, and even before Roman expansion in the 2nd century BC. We can say Proto-Albanian and Ancient Greek were spoken near, but not next to each other, even though Albania and Greece today share a border. There are clearly loanwords from Ancient Greek (in particular Doric dialect) to Proto-Albanian and vice-versa since 7th century BC according to this paper, but they weren’t in direct contact because these loanwords are not numerous enough. We can thus assume there were other languages in the Balkans spoken between Ancient Greek and Proto-Albanian, but those two were coexisting at least.

Linguists have studied in depth the Albanian language to figure out where was it originally spoken, an example would be Vladimir Orel A Coincise Historical Grammar of Albanian Language where he narrows done the place of origin by looking at the evolution of the language, comparing it with other IE languages, comparing regional/areal features that affect languages when they are spoken near each other even if they are not related genetically etc. and they generally fluctuate between these three theories: Illyrian, Dacian, Thracian.

Notice, I used “under this name” for Albanians before. That’s because the terms “Albania”, “Albanian” weren’t always the name Albanians were known as. That’s just the name they called themselves from at least 11 century to the 17th one. It generally accepted it comes from a city that in Greek was known as Albanopolis where a people called Albani lived, recorded in the 2nd century AD. Post-17th century, Albanians began to call themselves Shqiptarë, the language shqip, and the country Shqipëria. The point is, even though we don’t have enough information on Albanians, even though we cannot connect them with an ancient Balkan group of people with certanity, we still have enough to say they have been a distinct group of people for as long as we can tell. Distinct from Greeks, distinct from Romans, distinct from Slavs.

So, they didn’t become distinct because of the Ottoman Empire, though since culture and people always evolve, the Ottoman Empire most certainly affected them too. Robert Elsie’s The Tribes of Albania: History, Society and Culture can answer your question on both clans and influence of Ottoman Empire: Albanian clans are a direct result of the fall of feudalism in the region caused by the expansion Ottoman Empire, not some ancient culture preserved to modern day; it’s a reorganization of society fell. So, yes, the Ottoman Empire did affect Albanian culture in making them “different.”