Apple first brought its music service to Echo owners in the U.S. back in December 2018, and now it has expanded that availability to Fire TV. However, if you've haven't enabled the skill yet, open the Alexa app or online.
Sign in to your Apple Music account with your Apple ID. It was only in December that Apple Music came to Echo devices, and with Apple's rumoured video streaming service due to launch, we suspect the company will want to continue opening up its services to other platforms. Apparently, Apple is also looking to finalize a deal with Roku, a US manufacturer of digital media players that allow access to internet streamed audio or video services through TVs.
Apple's on the offensive with services right now.
You can choose Apple Music as the default music provider for Alexa, if desired.
Echo device owners who already have the Apple Music skill enabled and set up don't need to take any additional steps to access the content through their Fire TV.
Same reason iTunes is coming to Samsung televisions, and Airplay 2 to LG, Sony, and Vizio sets.
Earlier this year, Apple Music briefly appeared in the Google Home app (though it could not be used), but was quickly removed.