[steckerbrett]

Most microcontrollers have extremely low power sleep states, maintaining interrupts on external inputs often only draws the smallest amounts of power. When you're designing for low power electronics like this you often run into the problem of the self discharge of the battery completely dwarfing any consumption your microcontroller has. I'm reasonably confident they chose a lithium AAA simply because they have very low self discharge, and a AAA over something like a coin cell due to the rather enormous amount of power needed to transmit 802.11.

I get the feeling that this device is pretty tentative for Amazon, they haven't gone to huge lengths to keep the price of the build down. In particular there's also a connector on the bottom right of the PCB that looks a lot like it could be for a small screen or some other peripheral.