Jesus said they were not Jews so they were Gentiles. Jesus said they called themselves Jews and were organized so somehow the lie that they were Jews was current and he implies it was easily exposed. This implies they were physically uncircumcised as well as not in him spiritually. Ignatius says that the schism in exactly that city included Gentiles who preached Judaism; that's Judaizing, which usually includes circumcision but in this case got an exception. That would be typical as being one of the Simonian-to-Valentinian movements of the day. Ignatius said there were also Jews preaching Christianity without Christ; those are unbelieving circumcised Jews, which are not the ones Jesus speaks of, because he speaks of those in the church and not those outside. So the synagogue of satan were Gentile church members who called themselves Jews and counted themselves in the church but rejected the leaders' authority and excluded real Christians from their nominal "Christ" meetings which they called a "synagogue". Jesus said they would worship at the church's feet, and Ignatius implies that some of them did repent and accept his teaching, while others were baffled even as they continued arguing with him. Therefore the "synagogue" did not remain after the providential preaching and lecturing of Ignatius. That's the story.

The synagogue of satan today, if allegory means anything, is people in the church who do not believe it and who proclaim it to need a direction and organization that they cannot offer in truth.

If you call yourself an American Christian patriot on X and you're from Bangladesh, you're a patriot of satan by Jesus's metaphor, pretty simple.

When we get to the church being influenced by outsiders importing porn and pharmacy and transhumanism Jesus calls that the depths of satan instead. Different enemy attack.