The article makes the assumption that the direction was from Rome to Vietnam and not the other way around, yet provides no hard evidence. The far more plausible explanation is that it developed in both places independently given that both places have plenty of fish and warm weather, and at some point, during the silk road era, the two independent developments linked up.
There's nothing far more plausible about your explanation. If anything, two events occurring is less likely than one, given that plenty of other cultures didn't create fish sauce and yet had an abundance of fish nearby.
The evidence is that (according to another book, The Noma Guide to Fermentation) there is no known mention of fish sauces in south/east Asia before 7th century AD, while in the West it's well documented starting in the 5th century BC. The Noma Guide to Fermentation is however not an history book and is not written by historians, and anyway that's not hard evidence.
