Mechanical meters measure true power. They integrate the instantaneous product of current and voltage.
Quoting
https://en.m.wikipedia.org/wiki/Electricity_meter :
The electromechanical
induction meter operates by counting the revolutions of a non-magnetic, but electrically conductive, metal disc which is made to rotate at a speed proportional to the power passing through the meter. The number of revolutions is thus proportional to the energy usage.
The disc is acted upon by two sets of
coils, which form, in effect, a two phase
induction motor. One coil is connected in such a way that it produces a
magnetic flux in proportion to the
voltage and the other produces a magnetic flux in proportion to the
current. The field of the voltage coil is delayed by 90 degrees, due to the coil's inductive nature, and calibrated using a lag coil.
[16] This produces
eddy currents in the disc and the effect is such that a
force is exerted on the disc in proportion to the product of the instantaneous current, voltage and phase angle (
power factor) between them. A
permanent magnet acts as an
eddy current brake, exerting an opposing force proportional to the
speed of rotation of the disc. The equilibrium between these two opposing forces results in the disc rotating at a speed
proportional to the power or rate of energy usage. The disc drives a register mechanism which counts revolutions, much like the
odometer in a car, in order to render a measurement of the total energy used.