The way I described it was oversimplified. Technically, it's more like your credit card issuer issues a new card with the same number and installs it on the chip in the accessory.

To be able to do it, you have to authenticate with your card issuer in a mobile app, similar to how you might when setting up Android Pay or Apple Pay. The mobile app then uses your phone as a bridge between the issuer and the NFC chip in the accessory so the relevant data can be written in a secure way.