A serial number is the identifier of one specific pass. Where the pass type identifier names the program, the serial number names the individual pass inside it: together they form the full address of the pass in one member's wallet, and everything that touches a single pass, updates, registrations, support lookups, uses the pair.
The issuing platform chooses the serial number when the pass is created, and it never changes for the life of the pass. Wallets enforce uniqueness on the pair: a device will not add a second pass with the same pass type identifier and serial number, which is what prevents duplicates when a member taps an add-to-wallet link twice.
The serial number is what the update machinery speaks in. When a device checks in after a push update, it asks which serial numbers changed and fetches each one; when a member contacts support about their pass, the serial number is how one pass is found among thousands. It appears in device registration records for the same reason: each registration ties a device to a serial number.
It also matters at the exits. In a pass migration, serial numbers are part of what must be exported: the receiving platform has to keep addressing the same passes, or every member's pass goes quiet. A serial number is an identifier, not a secret, but a program should still treat the full list as its member roster, because that is effectively what it is.