I don't think it is quite accurate to say there were "brands" as such because that implies a standardization and mass production that only existed in embryonic form. There were, however, general social perceptions of quality for certain products associated with certain regions and at times this does feel "brand" like, but it should be kept in mind that this is at best a metaphor. Before giving a couple examples, the best way I can think to conceptualize this is the difference in how beer is sold and marketed vs how wine is--the difference between ordering a Dogfishhead 60 Minute IPA vs a glass of 2018 Chianti, if that makes sense.

I brought up wine in part because one of the closest examples we have to a "brand" are amphorae recovered from Pompeii stamped "VESUVINUM" which is both perfectly in line with Roman abbreviation and a pretty solid little marketing pun (Vesu vinum--Vesuvius) that has been enthusiastically adopted by a handful of local vintners today. It is highly unlikely that this was actually anything like a modern "brand" but it did play on contemporary perceptions that Campania produced quality wine. And in general wine, much like today, had common associations of quality with particular regions and years (the so called "Opimian vintage" of Falernian wine named after the eponymous consul of the year, had a certain legendary literary status). It was not just wine, however, iron from Noricum, in modern Switzerland, is a sort of natural steel and was highly valued, particularly by military quartermasters.

There were also famous craftsmen and workshops. For example, bronze medical instruments and other precision tools stamped with the name "Agathangelos" have been found in numerous sites across the western empire, and likely refer to the owner or master craftsmen of a particular workshop which was known for quality. This is not quite a brand in the way that, say, DeWalt is today, it does point to an empire wide network of information sharing regarding "product quality" that is somewhat "brand like".

So broadly speaking there was probably not anything really like "big brands" but there were people in Syria who knew they liked wine grown from a particular region in Italy, and there were doctors in London who were happy to have medical instruments from a particular workshop.