Wallet not working on iPhone 5s

Who knows exactly why, but...

Devices compatible with Apple Pay

Learn which iPhone, iPad, Apple Watch, and Mac models are compatible with Apple Pay and where you can use them for your purchases and payments.

Compatible iPhone models

• iPhone models with Face ID
• iPhone models with Touch ID, except iPhone 5s

To see if your iPhone supports Apple Pay, identify your iPhone model.

Devices compatible with Apple Pay - Apple Support

Wallet has apparently been around since iOS 6 but was known as Passbook.

If you can't add cards to Wallet to use with Apple Pay --> https://support.apple.com/HT205630