It appears that this system turns tapwater into hydrogen and oxygen by electrolysis.

I'm not a chemist, but I messed around with electrolysis decades ago when I was a teen who was into chemistry sets.

I am not convinced that tapwater is anywhere near pure enough that this system can work without frequently changing electrodes and removing gunge, even with some filtering.

Electrolysis does not just split the water molecules, there is a chemical reaction with whatever else is dissolved in the water.