Can an L293D motor driver shield do microstepping?
It would be more accurate to say that it can be used as part of a microstepping system, though it's probably not a good choice for any use today.

The L293D is merely an (effectively obsolete implementation of an) H-bridge, not a stepper motor sequencer, hence it is no more capable of autonomously driving a motor in full or half step mode than it is of microstepping - it can only do any of these when it is controlled by a sequencer, such as an L297 chip, or software running on an Arduino's processor.

Microstepping requires partially actuating the coils to interpolate between the angular positions which would be achieved by activating each in turn fully.  It can be thought of as a proportional extension of how half-stepping uses actuating both coils at the same power to achieve a halfway in between position, only now the coils are actuated at different power levels to achieve finer gradations of position, though achieving accurate interpolation requires care as the relationship is not linear.

Modern power electronics no longer use variable analog drivers, as these convert the unused power to heat.  Instead, the goal is to use switches which come as close as possible to only ever being fully on or fully off.  Reduced driver power is achieved by pulse width modulating the switch, varying the average amount of power delivered to the load over a short period of time.  In a stepper motor setting, this on-off modulation can be done to achieve microstepping and or also to achieve current regulation when a many-times-rated-voltage supply is used to allow overcoming winding inductance at high step rates. 

The L293D (and its larger cousin the L298) are quite inefficient bipolar transistor bridges (they aren't very good switches compared to MOSFETs), and should probably not be chosen for new designs today.  They are, however, capable of being used to microstep a motor when actuated by a sequencer which pulse width modulates the phase signals. 
Adafruit, who appear to be the original creators of this shield design, have in fact published microstepping driver code for their L293D board on github. 

https://github.com/adafruit/Adafruit-Motor-Shield-library

Given that the ebay seller you link to as the source of the board is using Adafruit's own schematic on their product page, it's a bit unclear why your software is differentiating between this board vs an Adafruit one, unless by the the Adafruit board you mean their V2 shield.  The V2 design uses a more efficient TB6612FNG FET bridge, but of more relevance to microstepping offloads the sequencing task from Arduino software to a PCA9685 chip which is apparently capable of microstepping (as well as simpler modes) by itself.  Depending on the computational load and/or your  level of determination, you may be able to get the ATmega microstepping the L293D in the context of your overall project.