'Syrian' was most likely used as a generic term for people from the eastern Mediterranean, so Gregory may not have been referring to Syriac, though it is certainly the most likely choice, given its status as the lingua franca of the Near East at this point; the alternative is of course Greek.

There are only a few mentions of 'Syrians' in sixth-century Gaul unfortunately, most of the references (all relatively vague and not very informative, like the section you quoted) being from the various works of Gregory of Tours. As such, it is difficult to say very much about them and very easy to go too far beyond what the evidence says. Some were described as merchants, such as Eusebius of Paris, who according to Gregory of Tours bribed his way into acquiring the bishopric; it would be too far I think to say that he was an immigrant, he could just have been a merchant in the right place at the right time. Then again, Gregory also said that this dastardly Eusebius promptly replaced the episcopal household with Syrians, so there was probably a decent-sized group of Syrians in the city already.

In his other works, Gregory mentioned that he had a Syrian assistant who helped him translate a Greek hagiographical account into Latin, and the historian certainly had contact with those who travelled from the east, as he was aware of interesting tidbits from the other side of the Mediterranean, such as events within the imperial palace in Constantinople (which perhaps came from Syrian merchants/diplomats, who leaned towards miaphysitism and would thus explain Gregory's relatively even treatment of these 'heretics'). The most interesting source Gregory used was his personal conversations with an Armenian bishop who had once been captured by Persians, which just goes to show how people can still travel widely in this period. Elsewhere, we have mentions of Syrian merchants from the earlier Life of Genevieve and a Syrian from the later Life of Columbanus. I think that's it in terms of the literary evidence (though if anyone know of more, please let me know!), so it really doesn't provide us with anything particularly concrete.

It is however important to not to assume that every Syrian was a merchant, since in Gregory of Tours only two Syrians were described as such. We simply don't know what the Syrians did for their livelihoods nor why they were in Gaul, though trade with the east did continue in this period, especially as easterners dominated the Mediterranean economy at this point; how else would the plague arrive at Marseilles a few decades earlier? Who else would provide wine from Gaza for the Frankish elite to enjoy? Beyond that, I think I can only speculate about Gregory' (or other Franks') attitude to these Syrians. Walter Goffart argued in this article that:

No longer Roman but not yet a Frank, he [Gregory] found in his faith and its principles of conduct a position that was adequate for a portrayal of his surroundings - a world in which all men were neither insiders nor outsiders but merely potential citizens in God's kingdom.

I'm undecided about this issue myself, though it is true that Gregory evidently cared little about what being Syrian meant in terms of social identity or assimilation into wider Frankish society. I'm less informed about Frankish sources from the seventh century, but Goffart noted that a later author, pseudo-Fredegar, was much more 'Frankish' in outlook in his Chronicle. Then again, I believe there are very few (or perhaps zero) sources for Syrians in Francia after the Life of Columbanus I noted above; perhaps the Syrians were assimilated, perhaps they left in a massive exodus, or, most likely in my opinion, the sources simply didn't mention them, as each source was written with a purpose and no source I know of from the seventh century was intent on describing what Frankish society was actually like. Gregory's references to the Syrians dominate our understanding of this phenomenon, because he is basically our only source for most of Frankish history in the late sixth century, but as I have hopefully demonstrated, his account doesn't tell us all that much. We can therefore only speculate about the Syrians' occupations and identities, even though they must have lived through some fairly extraordinary events.