There was no such thing as an Ancient Roman Flag. What the Romans used instead of a flag were battle standards of various sorts and forms. Many public buildings had inscriptions on their facades. The letters S.P.Q.R. Senatus PopulusQue Romanus stood to tell everyone that what they had before them belonged to the Senate and the People of Rome.
Ships used no flag either and given the relative similarity of the vessels of different cultures such as Roman, Greek and Carthaginian distinction between them was by means of a statue at the front or indeed up on the mast.
The Romans did make extensive use of symbols and banners, particularly for military purposes to rally the troops. These had a particular and almost mystical significance so much so that when they were lost in failed military campaigns great efforts would be placed into retrieving them, preferably by winning them back.
The military insignia would be held by nominated individuals called “signifer” within each century, there were therefore two such men within each manipulus, 60 in a legion. In fact, before joining battle, each centurion would nominate a backup signifer to ensure there was always someone ready to hold the standard.
Initially it is likely that these insignia were not so much a flag but rather a bundle of straw tied to the top of a post. Gradually this evolved into a broad variety of symbols and shapes such as eagles, hands (manus) and other to denote the specific manipulus to which that standard belonged.