Live Photo was never added to the 6th gen iPod Touch.
You cannot take Live Photos with an iPod Touch 6th gen, but certain apps, you CAN view them, like the messages app and your email.
Live Photos is ONLY on newer iPhone 6S and newer iPad models (iPhone 72nd gen iPad Pro, 9.7 inch iPad Pro and iPad 5th generation)
Post your feedback about this here.
iPod Touch Feedback
Cognizant Apple employees read ALL feedback generated from all of the various feedback pages and transfers the data to the proper, responsible Apple teams and personnel , but NO Apple employees will ever respond with any type of direct, individual replies from the feedback you post.
Good Luck to You!