With the Signature SDK this is only possible after capturing a signature.
In order to obtain the unique ID before capturing a signature this can be done with the STU SDK for STUs or Wintab for display tablets (DTUs).
When using the Signature SDK, certain information about the capture event can be extracted from the AdditionalData property of the signature object. For more details see the list of Enum values for CaptData in the Signature SDK reference manual. Here is the relevant extract:
Information about the STU in use can be found by using the CaptDigitizer enum value (26). This returns the model of the pad along with the serial number in a string similar to the following:
STU;'STU-530';1.0.2;0;3GZQS00072
The last element of the string is the serial number (shown in bold above). This works with a 430, 530 and a 540 but it may not work with older STUs. It doesn't work with DTU devices.
The serial number can also be inspected by running Identify.exe (one of our STU test utilities).