Greetings, its_nobody.
I see that, when on a FaceTime call on your Mac, when you take Live Photo, it doesn't appear in your Photos app. I'm happy to help.
Take a Live Photo in FaceTime on Mac -- Make sure that both you and the other caller's settings are correct. This article provides the information and steps that you need.
Use safe mode to isolate issues with your Mac -- If everything was set up correctly, restart your Mac in safe mode and see if you're able to take a Live Photo in FaceTime. Afterwards, reboot normally and log into your own user account. Remember this when booting in safe mode: Graphics performance limited in macOS Recovery, safe mode, or Guest User mode
How to test an issue in another user account on your Mac -- If the issue is also happening in safe mode, use this article to create and test a new user account. This will demonstrate if the issue is isolated to your account, or if it's system wide. You'll need to activate FaceTime in this new user account to test.
Let me know how that goes, and take care!