There are 2 types of Audi serial number. It is simple to remove your Audi A3 radio. You will need a set of radio release keys (PC5-83). These are inexpensive tools, you can obtain them from Amazon.
Push the keys into the 4 holes in each corner of the radio, apply pressure and the radio will free from the dashboard. The radios label will be located on the radio's casing.
An Audi serial is 14 digits and starts "AUZ" for example AUZ1Z1F6412082. Please see image for reference.
If for any reason the label is unclear or ripped off, the AUZ serial will be compressed into the metal casing on the side.
With the Audi Navigation Plus system the serial can be displayed on screen rather than taking out the radio unit.