The last DC power system in NYC was shut down recently. It was used to run the elevators in some buildings built there at the turn of the last century. In the rest of the world commercial power is AC. Neither the current, not the voltage is ever constant in an AC system. Wherever did you get such quaint notions?
The meter motor has both a voltage coil and a current coil, thus the speed of the meter wheel is determined by the volts times amps (on an instantaneous basis so the speed of the wheel is an indication of the true power). This is integrated by the dials to indicate the power over time, giving energy in kWh.