(NOTE: This is not a definitive answer and is largely guesswork, but it might be useful for further answerers.)
At first I suspected it was some sort of weak spring or magnetic device which applied an opposing torque to the arm, so as to make the center reading unique. Searching Google, I found this explanation of the theory of a triple-beam balance offered by Ohaus, and it mentions that there is a magnetic dampener that couples to the free end of the arm that is used to stop the oscillation of the beam (presumably by the generation of opposing electrical eddy currents):
Unfortunately, the slide in question also specifically notes that "this resistance is to movement and not an attractive force thus no added torque is applied", so this is not the correct reason. I searched through the rest of the slides to see if there was any mention of an angle-dependent opposing force, but there didn't seem to be any.
However, it is obvious that there has to be an opposing torque that occurs when the arm moves up and down, as otherwise if the two arms were balanced, there would be a wide range of accessible angles which the arm would be stable in, whereas having worked with these sort of scales before I can say that there is a restoring force that depends on angle.
Possible leads of investigation: according to the same set of slides, the weighing platform makes contact with the balancing arm using a knife-edge bearing:
As a result, if the weighing arm angle is $\theta$, the torque $T_\theta$ provided by the weighing platform will "correctly" be
Otherwise, using a different sort of bearing could cause the load to be applied closer or further down the arm, destroying the $\cos(\theta)$ dependence.
However, there is no mention of how the sliding weights make contact with the arm. So I'd try to find out how the sliding weights make contact with the arm, as this could potentially be used to ensure that there is a net angular difference on how the two sides react to changes in angle. For example, if the right edge of the sliding weight makes contact with the balancing arm when the sliding-weight section is raised higher, it would cause the torque to decrease slower than the normal $\cos(\theta)$ dependence that the left-hand side experiences, and thus there would be a restoring force that drives the sliding-weight section downwards.
Or maybe the post which is connected to the weighing platform and extends downwards into the hole on the top of the balance is actually connected to a spring:
Unfortunately I don't have access to a mechanical scale at the moment (all mine are digital).
Honestly the easiest way would be to ask one of their engineers, but that'd sort of be cheating :).