Can I, Bi-Directional Level Shifter as Motor(3V) controller with Arduino Lilypad (3.3V)?
No.  You should not use a "level shifter" as a motor driver "instead of using transistors for each application"

Although your motors are relatively low current as such go (though may well draw more in some situations), they still require at least three times more current than what relatively high-current "logic" applications would require - and at least an order of magnitude beyond what typical modern logic needs on a steady-state basis.

Motors are motors; logic is logic.  For driving motors, you should be using at least light duty power electronics rated for the anticipated loads.  There are definitely IC drivers in existence with sufficient capability - but they are marketed as that, not as logic level shifters.

Ultimately, it is possible that some particular level shifting designs using discrete components might be made of parts with sufficient rating for a light duty motor-drive application - but that would be something you would determine only after a qualified review of their specific data sheets, rather than something you can guess from the "level shifter" functional description of a circuit.  Further, motor drives typically incorporate protection components, such as catch diodes and suppression capacitors.