My new iPad Mini forgets my fingerprints every time I restart it. Is there a fix for this?

The message shown indicates that Touch ID or Face ID is disabled, and the passcode or password is required, when the device reboots.

Once the passcode or password is entered once upon reboot, Touch ID or Face ID will resume working.

Which is what you are encountering.

This requirement is intended as a security feature.

Touch ID and Face ID are a shortcut for the passcode or password, and intentionally not as a replacement.

Touch ID and Face ID makes using a longer and more robust passcode or password much more practical, too.

But Touch ID and Face ID are not a replacement.

See the Apple Platform Security section When a device passcode or password is required for details on when and why an iPad or iPhone will prompt for the passcode or password. And on how Apple strives to keep the data contents secure against unauthorized access attempts.

Hello,

When any Apple device restarts it requires that you type in your device’s passcode in order to unlock the device. Once the device has been unlocked then Touch ID will function as designed.

If Touch ID isn't working on your iPhone or iPad is what Apple documents for troubleshooting Touch ID.

Once past that troubleshooting, you’re then headed for a discussion directly with the folks at Apple Support.