Now there's a snappy title. Just marshalling some previous thoughts, any comment welcome.
Reading Maurice's Strategikon, translation by George Dennis, Maurice advises that the heavy infantrymen of each 'arithmos or tagma' (units of 2-400) should have shields of the same colour (Book XII:B4), implying that by the end of the 6th Century in the Eastern empire larger units did not routinely have identical shields for a legion or other large unit.
However, they may have had different colours with the same device: For the main force of cavalry, the emphasis is understandably upon the flags, which are to have a specific field colour for each meros (division of 6-7,000), specific streamer colours for each moira (2-3,000) within the meros "so that each individual tagma may easily recognise its own standard. Other distinctive devices known to the soldiers should be imposed on the fields of the flags, so that they may easily be recognised according to meros, moira and tagma." (Book I:2) - this does imply a unique device for each tagma within the moira.
However: "In each meros the flags or standards of each tagma should be fairly small and easy to carry.... The only distinctive feature should be in their streamers. But the flags of the moirarchs should be larger and of different design." It goes on to say that the flags of the merarchs, lieutenant general and general should all be distinctive. (Book II:14) This may be read that the tagmas have a common design, but that 'only' might be merely emphasising that field colours were common within a meros.
In his paper on Byzantine Flags, it seems that Dennis describes how more streamers were used for larger units, up to eight. The devices included circles, squares and, particularly, crosses. I'd like to get hold of it to check it out.