Correct on what it is. But it’s a DC power supply. Think of a diode rectifier. That gives us around 180 V though...way too much. So if we use an SCR we can turn the diodes on as needed, like a lamp dimmer. That gives us control over the DC.
Water temperature, current/voltage, and air temperature are common signals that the control box uses to adjust the output. It is possible to tightly control chlorine levels with another sensor but they are expensive and need periodic calibration and replacement so only used on some large commercial pools. The system is so stable even most commercial pools don’t bother doing that.
Either way the DC is between the plates. But it is hardly isolated from the AC power line. So the DC negative and AC neutral are either equal or offset by a constant (half of the DC output) depending on if it’s a half or full wave bridge rectifier. My bet is on half wave.
Three possibilities I can think of. The DC negative in the cable is broke and ground/neutral is now the return. SCR shorted (typical failure mode) and inducing AC into water but surprised it didn’t blow fuses. Or bonding to control panel is missing so it is floating probably around 60 V.
Get your meter out and measure ground to water, etc., to narrow down. Check the chlorine generator manual on testing.