FaceTime Photos must be enabled on both devices to use this feature
This camera button is the FaceTime Live Photos feature.

When you’re on a FaceTime video call with one or more people, you can take a Live Photo to capture a moment from the call. Both of you receive a notification that the photo was taken, and the Live Photo goes right into your Photos library.
The other person on the call may need to set their FaceTime preferences to allow a Live Photo before you can take one.

iOS &gt; Settings &gt; FaceTime &gt; FaceTime Live Photos
In addition to the Settings toggle, there is a minimum OS requirement. I got the same error message when FaceTiming with an older macOS computer.

Note: To capture a Live Photo in a one-on-one call, both you and the other person on the call must be using the latest version of either macOS 10.13.6 or macOS Mojave 10.14.4 or later, iOS 12.1.4 or later, or iPadOS. To capture a Live Photo of a person in a group call, both you and the other person on the call must be using macOS Catalina or later, iOS 13 or later, or iPadOS.
https://support.apple.com/guide/facetime/take-a-live-photo-fctm0ce5cbbe/mac