To make your Amazon Wish List anonymous, you can do this under settings. Amazon Wish Lists are public by default but each list is customizable according to your preference.
Hover over “Account & Lists” and click “Wish List” > “List Settings” (top right corner) > next to the list click the drop-down menu under “Privacy” and click “Private” (“you can also choose the “Shared” option if you want to only share the list with certain people”).
To hide everything from your profile, hover over “Account & Lists” and click “Your Account” > “Profile” (under “Ordering and shopping preferences”) > click the “See what others see” link at the top of the page > if you want to edit, click “Edit your profile” (you can change your public name and other information like your email address, location, social media links, etc. here) > “Edit privacy settings.” To hide all information from your profile, check the “Hide all activity on your profile” box and “Save.” People can search for lists associated with your email address or name.