The Roman Empire did not have a national flag in the modern sense. It did, however, have two rough analogues, one formal and one functional. The formal analogue – alluded to in the question – were the vexilla (standards) of the legions. The functional analogue was the image of the emperor.
The vexilla were, famously, small square cloth squares attached to the crossbar of a standard. The only extant example (image below), now in Moscow, is made of red linen and about 50 cm square, with a fringe on the lower edge. Artistic representations from various other contexts illustrate the broad range of iconography on vexilla. Some, like the Moscow example, showed the goddess victory. Others depicted various animals. Others still simply bore the name of the unit. All were flag- like in appearance – but none, of course, were anything resembling a national / imperial flag.
The closest functional equivalent of a Roman national flag was the image of the reigning emperor, which represented the (notional) unity and power of the Empire. The imperial image was ubiquitous, as illustrated by a frequently-cited letter of Fronto to his pupil and friend Marcus Aurelius:
“You know how in all money-changer’s bureaus, booths, bookstalls, eaves, porches, windows, anywhere and everywhere there are likenesses of you exposed to view, badly enough painted most of them to be sure, and modeled or carved in a plain, not to say sorry, style of art, yet at the same time your likeness…” (4.12.4)
The emperor’s image (standardized, at least in theory, from official portraits) was the face of the Empire. It punctuated the public places of cities throughout the provinces. It peered out from hundreds of thousands of coins. It was venerated, worshiped, and occasionally even savored (several molds have been discovered for making cakes stamped with the imperial image). The Roman Empire never came closer to a national/imperial symbol – at least until it became Christian, and the Christogram assumed something like that function.