Why does is Greece seen as part of Western Europe, when, culturally, historically, and genetically, it has closer ties to southeastern Europe and the Middle East?

The short answer is that it usually isn’t, not so explicitly.

For the longer answer, we first we have to ask: what is Eastern Europe? The division between ‘Western Europe’ and ‘Eastern Europe’ - and possibly ‘Central Europe’ - is vague, varies by period, country and even person, and has changed depending on various factors.

One major historical division was that essentially inherited from the division of the Roman Empire between East and West, which led most relevantly to the religious divide between Greek and Latin rite churches and later (with a little overlap) Orthodox and Roman Catholic churches. Most - but far from all - of Eastern Europe has an orthodox history, and modern Greece of course ties in with that.

However, exceptions abound; the Baltic states have a very mixed history here, with influence from Catholic and later Protestant Scandinavia, Orthodox Russia, and Catholic Poland. Poland, Czechia, Slovakia, Slovenia and Croatia have historically largely been Catholic. Bosnia, Albania and Kosovo are Muslim exclaves from Ottoman influence.

Other attributes associated more with the Eastern European countries include a predominance of Slavic (or more broadly Balto-Slavic) national languages, this excludes Estonia (Estonian is Finnic), Albania (its own branch of Indo-European), Hungary (Ugric, within Uralic), Romania (Romance), and European Turkey (Turkic). Romanians are mostly Orthodox but speak a Latin language, while Hungarians are Catholic and speak a language descended from much further east.

This doesn’t address the question of whether the Caucasus falls within ‘Europe’, itself not entirely well-defined, though use of the Ural River a border has been increasing elsewhere.

Poland, Hungary, Czechia and Slovakia are often included in ‘Central Europe’, given their historical ties to the Holy Roman Empire. (Note Hungary has historically been neither Orthodox nor Slavic).

But the most predominant divide recognised recently in the English speaking world, and explaining why Finland is usually considered ‘Western’ while Estonia is usually considered ‘Eastern’, and Greece is considered to align as ‘Western Europe’… is the Cold War. Countries that were under communism were considered Eastern Europe, simple as that. Greece and even more so Cyprus are very far East and so are tend not to be described as explicitly ‘Western’ so often, but were not included in Eastern Europe geopolitically. The reason that is the case is that Stalin and rhe Western Allies agreed that Greece would not fall within the Soviet sphere post-war, and British troops arrived in Greece in 1945. Despite this, arguably the first Cold War proxy war was between the communists in EAM-ELAS and its allies on one side and the Greek government on the other in the 1943-1949 Greek Civil War, where the government eventually defeated the communists with British and then American help.

There are still economic, cultural and infrastructural holdovers from that period, so the alignment has approximately stuck in Anglophone minds, despite efforts to popularise ‘Central Europe’ as a third category.