Is the Face ID fix free of charge?

That would depend on why it's broken. If it is a manufacturing defect and your phone is still under warranty, then yes, it will be a free fix.

Make an appointment at the Genius Bar to have it diagnosed.

Genius Bar Reservation and Apple Support Options - Apple

“Face ID: Is the Face ID fix free of charge?”

Perhaps.

Troubleshoot Face ID, yourself…

Overview these two links:

• If an Alert Says Face ID has been Disabled on your iPhone or iPad Pro - Apple Support

• If Face ID isn't Working on your iPhone or iPad Pro - Apple Support

Follow these steps, testing after each step:

• Face ID not working.
• Software Update.
• Turn iPhone on or off.
• Force Restart iPhone.

If that didn’t work, then it could be a hardware issue.

According to Apple:

Using diagnostics available in iOS 14.6 or later, Apple has detected an issue with your TrueDepth camera that might cause it to not function as expected. The TrueDepth camera contains lasers, which might be disabled for safety reasons if the device malfunctions or is damaged. This won't prevent you from using your iPhone or iPad, but until your iPhone or iPad is updated or serviced, you might not be able to use Face ID, Animoji, Memoji, and some other features that use the front camera.

As a result, I recommend service.

Service Checklist:

• iPhone service pricing.
• Check AppleCare Coverage.
• Get your iPhone ready for service.

Click ▶︎ Start a Service Request. ⚛️

Need more help?

• Chat now or call 1-800-APLCARE.
• International Support.

Click ▶︎ Check Repair Status. 🔧