Hello The_aze,
To help resolve your issue with the Touch ID on your MacBook Air, follow the steps from this article:
If Touch ID isn't working on your Mac - Apple Support
Try these steps
Try using Touch ID after each step:
1. Make sure that you have the latest version of macOS.
2. Make sure that your fingers and the Touch ID sensor are clean and dry.* Use a clean, lint-free cloth to wipe off any dirt or debris on the Touch ID sensor.
3. Your finger should cover the Touch ID sensor completely. While Touch ID is scanning, you shouldn't tap too quickly or move your finger around.
4. Choose Apple menu > System Preferences, click Users & Groups, then click the lock
to make changes. Click Login Options, then make sure that Automatic login is off.
5. Choose Apple menu > System Preferences, click Touch ID, then make sure that "Unlocking your Mac" or "iTunes Store, App Store & Apple Books" is on, and that you've added one or more fingerprints.
6. Try to add a different fingerprint.
7. For Magic Keyboard with Touch ID, turn your keyboard off and on, then connect your keyboard to your Mac with the USB to Lightning cable. After you turn Magic Keyboard off then on, Touch ID may be unavailable for up to 10 seconds.
8. Restart your Mac.

Thanks for using the Apple Support Communities, and have a great day!