Why didn't the photos I took on FaceTime call did not get saved to my photo album?

Ok, did you get the Notification that the photo was taken. Both of you should have received the Notification. Here are 3 tips:

• Make sure your device storage is not full
• Make sure both of you have that setting turned on to Allow the photo to be taken. If you did not want you have your photo taken, you could switch it off.
• Restart your iPhone. Some users have found that it works after restarting.

Did you take the photo using the provided method here:

Take a Live Photo in FaceTime on iPhone – Apple Support (MY)

Or, did you attempt to do a screen shot?