I would expect 10 to 20 mA average and 40 to 80 mA peak segment current for a 2" display. It has multiple LED chips (e.g. 2-4) in series connection, in so far a higher supply voltage than 5 V is required. Due to the forward voltage, the segments can be possibly still driven by a 5V open drain circuit.
The problem is however that 8051 GPIO output currents are far too low to drive the segments at full brightness. You didn't mention any resistor values, I assume that you are respecting 8051 maximum ratings, e.g. 15 mA for bank 3 which limits the segment peak current to about 2 mA.
The current strength of the digit drivers is sufficient for GPIO driven segments, but not if the display shall be operated at full brightness.