This is more a linguistics question than a historical one exactly, but I'll give it a shot. As you note, the word "synagogue" is Greek. And just as you say, Greek was a common language of the Eastern Mediterranean in ancient times. There were communities of Greek-speaking Jews in ancient times, too. From there the word passed into Latin and other European languages. It's used many times in the Christian bible, which would keep the term in currency even in parts of Europe without Jewish communities.

However, many Jewish communities didn't necessarily use the term "synagogue" at all. Ancient Jewish texts in Hebrew/Aramaic tend to use בית־הכנסת (roughly meaning "assembly place"). This is the term in Jewish texts in Hebrew and Modern Hebrew. Derivatives of this term also were widely used in Middle Eastern Jewish communities, such as "Kanis". The first attestation of written Yiddish uses בית־כנסת as well. Ashkenazi communities generally used "shul", which is cognate with "school" in English (and also means "school" in Yiddish).

Jews from Spain and Portugal generally did use a derivative of "synagogue", such as "esnoga", though. And of course it's used in English. While many Jewish religious terms are borrowed directly from Hebrew, many aren't, so it's not that unusual. The Yiddish "bentsh", meaning "bless", still used widely in Ashkenazic Jewish communities, is ultimately Latin. Yiddish words for praying aren't Hebrew either (the most common, "daven", has no obvious etymology--but Western Yiddish "orn" is Latin). The now-obscure English word "porge" for removing forbidden fats from meat is from Judeo-Spanish (nowadays various forms of the Hebrew "nikkur" are usually used), related to "purge".

In general, though, the trend over the past century or so has been towards more Hebrew/Yiddish. It's pretty unusual to refer to Jewish holidays by translated names, for example, but it wasn't 100 years ago. As noted above "nikkur" has replaced "porging". "Synagogue" has mostly resisted this trend. I say "mostly" because using "shul" in English has gotten more common over time (at least, that's my perception). I don't think there's an obvious answer why. I suspect that the reason is that the term is neutral. Reform Jews (and many Conservative Jews) refer to the synagogue as "Temple", a trend that began in 19th century German Reform Judaism. While people who use the term are usually not consciously choosing it for theological reasons, it does have theological undertones for many Jews and is often thought of as Reform-specific (it's also used by Italian Jews, but not for that reason and without that connotation--the Great Synagogue of Rome is "Tempio Maggiore" in Italian). Terms like "shul" are Ashkenazi specific, may be thought of as suggesting an Orthodox synagogue, and aren't necessarily familiar to all English-speaking Jews. Plus "synagogue" is much more entrenched as an English term than, say, the English names of Jewish holidays.

So, "synagogue" sticks around in English as the default. Jews among themselves might use "temple" or "shul" or "shil" (the same as shul but in a different Yiddish dialect) or "bet knesset", but "synagogue" still has enough currency to stick around, and the Hebrew or Yiddish alternatives aren't widely enough used or known to replace it.

Well, Jews don't necessarily call it a synagogue! The Hebrew term used is beit knesset (בית כנסת), the Yiddish term is shul, and many Jews call it a temple. That said, the word synagogue has made its way into the English language (and other languages) for a few different reasons, and one of them is in fact that Greek was a dominant language at the time that the synagogue concept is said to have originated (there is scholarly debate as to the exact time line).

The term synagogue for a Jewish locus of worship (in slightly different form I believe, but I'm not all that familiar with Greek to really grasp the difference) originated when the scholars who translated the Hebrew Bible into Greek (the Septuagint) in the 3rd century BCE needed a word for the Hebrew edah (עדה). In English that's often rendered as "congregation" or "gathering" and the word that the Septuagint translators chose was synagogue. Another Greek term that was in use at around the same time in Egypt was proseuchē., though that term wasn't exclusively used for Jewish worship. In addition to the term beit knesset, other Hebrew terms were also used.

The reason why there were so many terms is because the concept of the synagogue was still in flux- or, at least, modern day scholars are having trouble nailing it down. As you might have seen from the fact that "synagogue" is a translation of the word "edah," it's less about the place where Jews gathered and more about the fact that they were gathering. (The term "proseuchē," on the other hand, was more clearly about the place.) But the function of these gatherings and gathering places is still unclear, and was definitely something that would come to change over time, particularly after the destruction of the Second Temple in 70 CE. After that, increased numbers of synagogues popped up to replace the larger national center of worship, and the worship that was conducted there became more standardized over time.

Once we reach this point of standardization of rite and ritual, much of the writing about the synagogue and its laws and standards ends up being in Hebrew and Aramaic, and the term synagogue is used less among the Jews who use them (unless they live in areas where Greek is still spoken). However, the Septuagint, which originated the use of the term, then became the source text for the Christian Old Testament- spreading use of the term among Christians. This becomes relevant when Jews become a minority in a Christian majority population, and soon the word "synagogue" in various forms has become less a term for a particular building or gathering and more as a conceptualization of the Jews themselves. "Synagoga and Ecclesia" or "Synagogue and Church" are seen as opposing forces as personified in a set of statues- in these statues, Ecclesia is depicted as a proud and upright woman and Synagoga is depicted as a downtrodden and blindfolded one. Even as Jews were using alternate terms for their own houses of worship, the Christian term came to dominate the language and culture of the Western world.

In short, synagogues were developed when Greek was the language in use for much of the Jewish population. Hebrew despite being revived in modern times was not the common language of many during the time period in which we start to see synagogues. The language that was predominantly in use was Greek. The Hebrew word for the same place is Beit Knesset (בית כנסת) meaning house of assembly.

Jews themselves do have several names for them, depending on their location in the diaspora. Ashkenazi Jews (those in Europe) would cal lit a Schul which comes from the German word for school 'Schule'. Sephardic and Rominate Jews which were originally in Spain, Portugal, and Rome and around the Mediterranean area would have called it a Khal, which comes from the Hebrew word Qahal (also transliterated Kahal) קהל and esnoga or sinogoga Persian Jews use a term from Aramaic kensea.

The earliest inscription we have about a Jewish study house is found in Egypt during the 3rd Century BCE and the inscription is in Greek but the word used is proseuche or place of prayer. There is also an inscription dating from near the end of the second century CE in Jerusalem in the City of David which uses the term synagogue. During this period Jerusalem was under Hellenistic (Greek) influence and control and the language in use was Greek. Romans categorized these institutions as collegia. Josephus in his writings in the late 70 CE used the term synagogue which he clearly notes is a place that houses Torah scrolls and is used for public gatherings. In the Christian texts, there are numerous greek terms used for places of worship with one of them being a synagogue.

The Sardis synagogue in modern-day western Turkey completed in the 3rd Century BCE has inscriptions in both Greek and Hebrew, Greek is used for donors and dedications throughout the Synagogue. This had locations for Torah scrolls and would have been a gathering place for the community. The inscriptions inside have been badly damaged and many are only fragmentary, the Greek inscriptions for donors follow a pattern of: "[Name], [Some Sort fo Title], and I have fulfilled (or executed) a vow" vow is used for general floor mosaics and specific items are referred to otherwise as in executed the ornament of the skoutlosis (Torah holder). The surviving Hebrew inscriptions are fewer in number and some are solely the word vow and another is the word Shalom (peace), and in another instance an inscription which is possibly graffiti which reads "I Shemaryah son of (E)lijah I wrote this". There isn't a specific word used in these surviving inscriptions for the name of the building, but there is clear evidence of the use of Greek.

Despite these early examples, the exact origins of synagogues are somewhat uncertain and debated, and there are several candidates for being the origin of them. However, it is important to note that the absolute need for them did not come about until after the destruction of the Second Temple in Jerusalem by the Romans in 70 CE.

When the temple was present the Jewish religion was primarily focused on being a temple cult with sacrifices and rituals performed in the temple at its center. After the destruction of the temple, this shifted to a Rabbinic influence with prayer and study at its center. This shift created what is now known as Rabbinic Judaism. This would have been accomplished by the group called the Pharisees which are first mentioned by Josephus as having existed near 150 BCE and arose in response to the Hellenistic groups running the Temple. It is also important to note that prior to this Judaism was divided up into a few different groups with different ideas on how things should go. After Rabbinic Judaism, that pretty much became the standard with a few minor exceptions (Samaritans and, Karaites most notably). But even this need is debated, as noted above we see many examples of synagogues in use prior to this. But the role of the refocus on prayer meant that a communal space was now a necessity everywhere.

In the Babylonian Talmud, codified around 500 CE, the term Beit Knesset is used as it is in Maimonides writing the Mishne Torah, which was completed between 1170-1180 CE, for example here בְּבָתֵּי כְּנֵסִיּוֹת is one example of him using it, the translation here is "inside the Beit Knesset". Maimonides was a native Arabic speaker since he was initially in Islamic Spain prior to fleeing to Morocco. We see the term synagogue in other places for example the Shulchan Aruch, a codification of Jewish Law written in the 1500s.

Synagogue is not the only Greek loan word used the term Sanhedrin, which were the councils (Great and lower) or assemblies (courts) for Jewish law in post-exile times is also Greek. There are many other words of lesser importance, including a word used in religious texts meaning 'overflowed', for example, codified in Babylon that are also Greek in origin there are also items in Latin as well. These texts are largely written in Hebrew and Aramaic, the Aramaic consisting of 2 variants from different locations.

So overall we see many languages that would have influenced word choices even in religious contexts in Judaism of this era.

I'm going to write some sources here since I had to pull inscriptions from JSTOR so I don't lose them:

The Hebrew Inscriptions from Sardis, Frank Moore Cross The Harvard Theological Review Vol. 95, No. 1 (Jan., 2002), pp. 3-19

The Greek Inscriptions of the Sardis Synagogue, John H. Kroll The Harvard Theological Review Vol. 94, No. 1, The Greek Inscriptions of the Sardis Synagogue (Jan., 2001), pp. 5-55

Review: Greek and Latin Loan-Words in Talmud, Midrash and Targum Reviewed Work: Griechische und Lateinische Lehnwörter in Talmud, Midrasch und Targum by Samuel Krauss Review by: Marcus Jastrow

The American Journal of Semitic Languages and Literatures Vol. 15, No. 3 (Apr., 1899), pp. 184-186 (3 pages)

THE ORIGIN OF THE SYNAGOGUE: A RE-ASSESSMENT RACHEL HACHLILI Journal for the Study of Judaism in the Persian, Hellenistic, and Roman Period Vol. 28, No. 1 (1997), pp. 34-47

The Ancient Synagogue from Its Origins to 200 C.E. by Anders Runesson, Donald D Binder, Birger Olsson