Taking pictures with headphone jack It's a first generation Moto G running Android 5.0.2 and I'm trying to make this headphone-jack selfie stick work.

Tried with both Motorola Camera and Google Camera apps. Both take photos correctly when the volume button is pressed on the phone. Neither take a photo when the button on the selfie stick is pressed. I would consider the product defective, but Google Camera actually focuses when the button is pressed.

Is there any setting I'm missing, or a way to find what the selfie stick button is actually doing so I can re-map the action?
Check your volume sliders, it sounds [see what i did there? xD ] like
it's your notification volume that triggers the mechanism. At least that what i got when you said pushing on the volume button did the trick. 
Unlock/unlink your volume sliders if they're locked/linked, also if your rom has safe earphone volume feature, disable it.