Inspect your connector terminals both on the car and on the EVSE plug for any signs of dirt, debris, moisture, cracking, black sooty marks from arcing, etc. Check the AC voltage of your EVSE plug to be sure you are getting mains power on the correct lines. What is your AC charging voltage--is it from 220VAC, 50 Hz? Is it 2 legs of a 3-phase circuit? Maybe a phase circuit breaker has tripped one leg and your power is floating, i.e. there is no return path to complete the circuit. Be careful for sure until you figure this out.
Have you had any service work done recently on the vehicle, maybe in the rear engine compartment area?
One of the phase lines from the AC voltage coming into the car has a (hopefully high resistance) short-circuit connection to chassis. When you touch the car and ground at the same time, then you complete the circuit and get a shock.
So now use your meter on AC Amps~ [you will need to move the red meter lead to the A hole in the range that you are testing] and measure between chassis and ground to see how much current will leak thru. Start on the highest scale and work your way down to get a good reading. If you take more pictures, show your charging plug and the connection to the mains all the way to the car if possible.