Author Update:
Today, October 28, I upgraded my iPhone 12 from iOS 18.1 beta (22B5075a) to iOS 18.1 (22B83). However, the option to add travel cards still does not appear.
I tried using my old iPhone 11, which is on iOS 15.6.1 (19G82) and logged in with the same Apple ID as my iPhone 12. On this device, I was able to add travel cards. After adding the cards, I used the "Previous Cards" option on my iPhone 12 to add the new card I just added, since both devices share the same Apple ID.
For testing purposes, I then upgraded the iPhone 11 from iOS 15.6.1 (19G82) to iOS 18.1 (22B83). As a result, the option to add travel cards disappeared.
Summary: Pretty sure it is a bug in iOS 18.1.
- iPhone 11 iOS 15.6.1 (19G82) - Can add travel cards
- iPhone 11 iOS 18.1 (22B83) - No option to add travel cards
- iPhone 12 iOS 18.1 (22B83) - No option to add travel cards