Put a 1k resistor on the relay where the motor will go.
Then test again.
You will probably then see that there is 0v when off and 12v when on.
If that is the case, then all is fine, you can remove the meter.
My guess is that your meters high impedance isn't loading the system enough. And that is allowing the smallest current tracking across the board to be picked up by your meter.
The motors impedance will do the same thing as the 1k resistor. It will load down the circuit, and you should not have a problem.
High impedance comes in handy sometimes when I test mains. If I just want a quick confirmation that the circuit is live/dead, I hold one probe in my hand (touching the metal of the probe.), and touch the other probe on the terminals. The Live terminal normally gives me a reading of 100v or above (and this is through safety shoes), and the Netural terminal (or a dead terminal) gives me 0v...