The RMS had six joints. (It is shown here holding the Orbiter inspection boom.)
The rotational position of each joint was measured by internal 16-bit optical encoders giving a precision of 0.0055 deg.
However, since no mechanical device is perfect, corrections (known as "encoder biases") for the differences between "encoder zero" and "mechanical zero" were loaded into the flight software and indexed using the RMS serial number. The serial number was displayed on the main RMS Operations onboard computer page, SPEC 94.
The flight software used the joint angles, corrected by the encoder biases, to calculate the position and attitude of the RMS point of reference (POR) in use (for example, the tip of the arm).
If the wrong RMS serial number was entered, a bad set of encoder biases would be chosen, and the calculated position of the POR would be slightly different from its actual position. This graphic shows an error at the tip of the arm caused by a 0.25 degree bias in the shoulder yaw joint encoder.
The information loaded into the flight software was carefully checked for the real Orbiter and I am unaware of any issues caused by a serial number mismatch in flight. The crew would normally not need to interact with the serial number field on the display.
In training, however, the particular arm that was set up in the simulator had to match the one set up in the flight software, or problems would arise. Similarly, a changeout of an RMS on an Orbiter due to some problem late in the processing flow would result in the need for a data update to the flight software.