Tachometer air core meter - how does it operate?
A permanent magnet is affixed to the shaft and there are two coils at 90 degrees from each other, the current ratio between the two coils should be the sin and cos of the desired angle.
Think of a vector- the length is not so important but the angle is important. Photo from here:

Ideally, the needle will rotate to an angle \$\theta = tan^{-1}(i_x/i_y)\$ where ix and iy are the currents through the coils.
There are a number of ICs designed for driving automotive air-core gauge movements.
The voltage across each coil (not to ground) will give you a rough idea of the current, but the copper resistance varies with temperature and one coil will not be the same resistance as the other (same number of turns, but different resistance, since they can't occupy the same space).  If you add a resistor in series with each coil (an ohm or two) you should be able to read the currents accurately with a multimeter on the 200mV range.