Telephone line and cable line to the house are both shielded which would prevent any inductive coupling unless seriously damaged (and over very long lengths) but then the service would be affected. Plus, the conduits underground are separated as well as the wires on the poles so very little inductive coupling would occur even without shielding. But inductive coupling doesn't really explain the potential difference between the railing and pool water. That needs to be something in proximity to either one. I see this as being a very low probability.
For troubleshooting, I would first do a continuity test from the railing to the bonding wire at the pump, disconnected from all equipment at the pad, to see if it is bonded or not. You may need another long wire to accomplish this but the resistance should be negligible with a heavier gauge wire. I would also measure the resistance between the motor shaft and the bonding lug. That should be very close to zero.
Another measurement would be a voltage measurement, both AC and DC, between the bonding wire and the ground wire (not connected to one another) and the railing to ground and railing to bonding wire. A high DC voltage (48v on hook) could indicate a telephone wire issue while a high AC voltage could indicate a mains or landscape lighting issue.
A lot of what you said here has been said before.
I have asked a couple of times if the railing has been temporarily bonded, the bond can then be removed to facilitate testing.
The measurements I have seen have been from the railing to the deck and the railing to the water. The railing being the common factor. 4 volts RMS and 5 peak are a couple I remember.
I have also suggested a wire and continuity tester. Wire attached to the bonding lug on the motor and then carried around with the tester and testing all metal with in 5' around the pool to see what beeps and what does not. Including all metal in contact with the water.
There is also solar on the property that he has had looked into. I am still kinda stuck on this one as a possibility.
When the main breaker is shut off the voltage persists. When the power in the neighborhood goes down it goes away. So it's fairly local to the house, within the grid he is on. That's why I suggested turning off the main and lifting the neutral, then the utilities bonding. Remove the ground rod and the make sure the bonding jumper is intact and working.
One of the pictures the OP posted had voltage at 178Hz 3rd harmonic I also can't let that go either. Solar again has to convert from DC to AC in the inverter. We all know that silicone chopping up electricity will cause trouble including but not limited to the 3rd harmonic.
As has been stated if the voltage path(s) are going to be found or at least inside or outside the residence confirmed certain steps need to be done. That is up to the OP and his time table and what he is willing to take on or hire out.
At this point he or someone appointed by the OP should start at the beginning of the thread and list the tests that have been completed and the results then the ones that have not been conducted and list and prioritize them (this is be a sticky wicket as everyone will have a different idea of what is important). This will also help those that have been contributing and have not read the entire thread.
We have given him so much to do that it would take two experienced people a week to finish.