If you're using the Kindle reader app, you have to give it the ok because once you install the app, it registers the device with amazon. If you go to amazon - Manage Kindle, you'll see your phone listed.
All registered devices can access the same book you've purchased and it can synch which page you've read last. I have phone, PC, kindle registered and I believe some products you purchase/download have copyright and may only allow you to download the product to so many devices. I experienced that when I was moving from a Blackberry phone to Android. I had to remove my BB before I could download the book to my Android phone.
Just a side note; If you're worried about giving out to many permissions on your phone, watch out, android apps want all kinds of permissions.