Prime/Amazon has nothing to do with iCloud storage space. Having unlimited storage on Prime does not give you unlimited storage on iCloud. They are 2 completely different services. To store photos on your Amazon storage, you will need the Amazon Photos app and can follow the steps here:
https://www.amazon.com/gp/help/customer/display.html?nodeId=GLW9PXEWV7XMT8JC
Your iCloud storage can be used for photos, but you do not have to use iCloud Photos. To turn off iCloud Photos, follow the steps here:
How to turn off iCloud Photos - Apple Support
You can also use iCloud to back up data on your phone so you do not lose data if your phone breaks or you forget your passcode. There is not a way to back up your phone to Amazon and that is only done by backing up to a computer or to iCloud.
Back up iPhone - Apple Support
By default your photos will be in your backup file if you do not use iCloud Photos. If you are using iCloud to back up your phone, the photos will be included in there and if you have a lot of them, it would require more iCloud storage in order to back up the phone. Since you want to use Amazon to store your photos, then you would want to turn off photos to be backed up to iCloud so you would not require that extra space to be used in iCloud for photos. To prevent the Photo Library from being backed up to iCloud, go to Settings > [name on top] > iCloud > iCloud Backup > [name of device > Photo Library, and turn that switch off.
Now you are not using iCloud to save any of your photos and you will be responsible to save them to Amazon using the Amazon Photos app to prevent any data loss. Your photos will be gone if they are not saved to Amazon and your device get damaged or you forget your passcode.