EAN-13 is the thirteen-digit linear barcode used to identify retail products worldwide. It encodes a GS1 company prefix, an item reference and a single check digit, and it is the format behind the barcode on almost every packaged product sold outside North America, where the twelve-digit UPC-A is the regional equivalent.
In loyalty it matters because retail scanners are built around it. A till that reads EAN-13 all day reads it reliably, so a membership number issued in EAN-13 form scans with the same speed and the same hardware as a bag of coffee. Programs migrating from plastic cards often already have EAN-13 member numbers printed on the back of the card, and keeping that number lets the wallet pass drop into an existing point of sale without a change at the till.
Wallet support differs by platform. Google Wallet renders EAN-13 as a pass barcode; Apple Wallet's barcode formats are QR, PDF417, Aztec and Code 128, so an EAN-13 member number on an Apple Wallet pass is typically carried as Code 128 or as a QR code with the same digits. Where a program needs one identical symbology across both wallets, Code 128 and QR are the safe choices.
EAN-13 is numeric-only and fixed length, which is a constraint worth planning for: it cannot carry letters, prefixes or a rotating token. For static member numbers that is fine. For anything that needs to change per visit, see QR and barcode identification for the formats that can.