Yes, everything is setup exactly as mentioned above. There is only one difference in my setup. Which is in the sentence mentioned bellow:
An Amazon Echo, Amazon Fire TV, or Alexa-enabled Sonos speaker that you already set up with the Amazon Alexa app on your iOS or Android device.
The device I am using is a Bose Portable Smart Speaker which has Alexa built in. I have successfully setup Alexa, added the Apple Music Skill and also linked my Apple Id. I have an active Apple One plan (which has Apple Music Subscription) on my Apple Id.
Device link: https://www.boseapac.com/en_in/products/speakers/smart_home/bose-portable-home-speaker.html#v=bose_portable_home_speaker_black_in
FYI, Airplay and Bluetooth works fine, I am able to play songs from my iPhone. The only problem is Alexa cannot play songs through Apple Music on its own, but I does well with Spotify and other streaming services.
As Apple Music is not working on my Bose Speaker System I was thinking of unsubscribing Apple Music and buying a Spotify premium plan. Before doing that, I just wanted to confirm if there are any ways to solve it.
Thank you in advance for the help.