Unofficial groups in the Greco-Roman world that I (and others) typically call “associations” used a variety of terms to describe themselves. Some of the favourite Greek terms were synodos (“synod”), koinon, synergasia (“guild”), thiasos (“cult-society”), and mystai (“initiates”). Today, when people (including many scholars) hear the term synagogue or head-of-the-synagogue (archisynagogos) they tend to assume some Jewish group (or building) is in mind. However, the term synagogue (stemming from the Greek synagō, meaning to gather or bring together) was also used by other “pagan” associations and was not necessarily a sign of Jewish connections.
Thus, for instance, one monument from Apamea in Bithynia (northern Asia Minor / Turkey), which involves a group of men and women devotees (thiasitai and thiastides) honouring a priestess of Cybele (the Great Mother), mentions that the inscription was set up in the “synagogue” of Zeus (IApamBith 35). Across the Propontis in Perinthos-Herakleia in Thracia, there was an occupationally-based “synagogue of oar (or small-ware) dealers” that shows no sign of Jewish connections (IPerinthos 59 [first or second century]). At both Beroia and Hagios Mamas in Macedonia there were associations (devoted to Poseidon and a hero-god respectively) whose main leader was known as the head-of-the-synagogue (archisynagogos) (IMakedD 747 [second century]; SEG 27  267). And there are many other “pagan” cases where the chief leader of the group, as in some Jewish gatherings, was termed head-of-the-synagogue (e.g. NewDocs I 5; IG X.2 288-289; SEG 42  625).
Diaspora Jewish groups (including Jesus-devotees) shared more in common with “run-of-the-mill” associations of the Greco-Roman world than often acknowledged, and their “gatherings” would have been viewed as such by outsiders in some important respects.
To read more about associations in the Greco-Roman world, as well as their relevance to early Judaism and Christianity, go to the Associations in the Greco-Roman World site (under my websites above).