I search and it seems no one has brought this up but along with the Music Unlimited service amazon also has a Kindle Unlimited service for $10/month. I am still investigating to what limitations (like how long can you keep a book on your devices).
Here is a link:
https://www.amazon.com/kindle-dbs/hz/signup?_encoding=UTF8&ref_=ku_kc_rw_lp
Here is a link:
https://www.amazon.com/kindle-dbs/hz/signup?_encoding=UTF8&ref_=ku_kc_rw_lp