With the Bop It opened up, we can see the different components that make the game work:

The button used to detect a twist. Looks like instead of using a dedicated sensor to detect a twist, the toy connects the twist to a lever to push a button. Maybe this was done to save money, but this may not make the toy as durable.

The button used to detect a pull. Like the twist part, this also uses a button instead of a dedicated sensor.

The button that detects a bop. Since a bop is just a button press, this one doesn't need any additional fancy mechanical parts.

The speaker. It looks like it has a spring over it that makes "bopping it" as fun as it is.

The microcontroller. It's covered up by a glob top, so we can't actually tell what type of microcontroller it is unfortunately.