MrFish quoted the Service Manual as saying:
Combination meter outputs the turn signal sound with the integrated buzzer while blinking the turn signal indicator lamp according to the turn signal indicator lamp signal.
I dug a bit deeper into this. The "Combination meter" is pretty clearly a computer. It controls the display of the lower dash area, including the bubbles, the three sets of bars, the DTE (GOM), and all warning and indicator llights. The "integrated buzzer" must be a general purpose tone generator. It is used for warning chimes as well as the turn signal clicks. I would guess it is used for all sounds associated with the dash (power up, etc.). It is told what to do by CAN messages.
Other than muffling all sounds by covering the sound diaphragm, I suspect the only way to reduce the turn signal volume (something I would love to see) would be to modify the firmware that controls the Combination meter. As MrFish suggested, the speed of the click and flash is actually controlled by a different computer, the Body Control Module. Each click or flash appears to be a separate CAN message.