Hello TimPollard,
Thanks for that info and for choosing the Apple Support Communities. If we understand correctly, you are unable to use Touch ID on your Mac when making purchases, as it asks for you to authenticate using Face ID on your iPhone. To make sure this is enabled properly, let's have you double check Touch ID settings on your Mac:
"To change these settings, choose Apple menu 
> System Settings, click [your name] at the top of the sidebar, then click Media & Purchases on the right. If you don’t see your name at the top of the sidebar, click Sign in with your Apple ID to enter your Apple ID or to create one.
Option
Description
Account
Click Manage to view and update the Apple ID settings you use for purchases from the App Store, iTunes Store, Apple Books, Apple TV, and more.
Subscriptions
Click Manage to view and update your current subscriptions for Apple Music, Apple News, and more.
Use Touch ID for Purchases
Require Touch ID to complete purchases from the App Store, iTunes Store, Apple Books, and Apple TV.
Free Downloads
Require a password when downloading free content from the App Store, iTunes Store, Apple Books, and Apple TV. If you turn on Touch ID for Purchases, this option isn’t available and Touch ID is required for free downloads.
Purchases and In-App Purchases
Require a password when making purchases from the App Store, iTunes Store, Apple Books, and Apple TV, and when making in-app purchases. You can choose to require a password with every purchase, or only after a period of 15 minutes has elapsed since the last purchase. If you turn on Touch ID for Purchases, this option isn’t available and Touch ID is required for purchases and in-app purchases."
And that is from here: Change Media & Purchases settings for your Apple ID on Mac - Apple Support
If you are still having this issue after following those steps, please contact Apple directly so they can look into this with you further: Get Support
Cheers!
