Why did the designer use this motor?
I'd say this was purely to reduce cost. With an AC motor, the device can use a transformer to step down the voltage (probably a cheap wall-wart), but skip having a bridge rectifier to convert it to DC. Further, AC gear motors are cheaper:


  AC motors are generally smaller, less expensive, lighter and more rugged than DC motors. DC motors, on the other hand, are generally considered better for variable speed operations.


(From Sinotech.com)

Since this device doesn't require variable speed, the AC gear motor choice seems obvious.

It must have been less expensive to provide a battery clip for the LEDs than a few diodes for DC conversion. (Or maybe lighting it was an afterthought.)

As to why the motor isn't DC and runs off batteries, I expect this is something you'd leave on for a while, and small (presumably AA cells?) would not run a motor continuously for very long.