I hope you understand what a streaming service is, but if you and others do not know a brief explanation.
Streaming music is where each song, album or collection is held on a remote server and you select what you want played and the order in which you want it played. You then play the Playlist you have created or select a Playlist that others have created and you then listen to the music over an internet connection, no download takes place. Mainly these streaming services are via a subscription but some may offer limited free access.
Amazon Prime Music is a free streaming service, limited in quantity, where you do not buy the music you simply play it on your computer, tablet, Alexa device or phone.
If you buy some music you can download it and save it to wherever you want and iTunes would not be my first choice since I do not want it downloading so it only plays with iTunes. That said you do not need
Prime membership to buy and download music.
Then Amazon also have the subscription based service Prime Music Unlimited which is again not intended for downloads it is for streaming.
You also get Prime Video where there are thousands of shows, sports events (ATP Tennis), movies available to watch.