Did the Byzantines of the post 4th crusade Byzantine Empire(so 13th-15th centuries) really know their lineage? Did they understand how important Rome or even the earlier years of the Byzantine empire. Did the average citizen know some of the history of Rome, did they feel a real connection to it beyond sharing titles? This could be for all social classes.
To a certain extent you can believe that they did, why? Because the city was literally littered with monuments commemerating their past. Its hard to not look at the circus maximus and not wonder why its there. So yes, but whether its very in depth, who really knows. Over a 1000 years have passed since then.