Does the biometry backup with iTunes? I have an iPhone X with my FaceID registered. I back it up using iTunes.

If I restore this backup to another iPhone with FaceID support, will my registered FaceID be restored? Or do I have to register FaceID again?

I don't have multiple devices to test these scenarios on, and I couldn't find a definitive answer online.
FaceID and TouchID are stored locally only on the device in a Security Enclave and never sent across the network.

As Justin Pot writes in "What Is Apple’s “Secure Enclave”, And How Does It Protect My iPhone or Mac?" on How-To Geek:


  The Secure Enclave boots separately from the rest of your device. It runs its own microkernel, which is not directly accessible by your operating system or any programs running on your device. There’s 4MB of flashable storage, which is used exclusively to store 256-bit elliptic curve private keys. These keys are unique to your device, and are never synced to the cloud or even directly seen by your device’s primary operating system. Instead, the system asks the Secure Enclave to decrypt information using the keys.


So, your FaceID (or TouchID for that matter) is tied to a specific device and can not be restored on a different device from a backup.

You can read more about it in the Wikipedia article on FaceID.