The 6th generation iPod Touch cannot install iOS 13 or later, so there is nothing you can do about that.
Apple's message that your iPod has the latest version of iOS installed is misleading (as you are now discovering). The message should say that your iPod has the latest version for your model.
The only options that I can think of, that would enable your use of Amazon Music, are to:
- buy as second hand 7th generation iPod Touch (whihc can use iOS 15, but not iOS 16)
- a recent iPhone
- an Android device (that is recent enough to enable use of the Amazon App (I believe that is Android 5.0 or later)