I've just checked it out, and apparently BookLikes handles the Kindle integration via a member's Twitter account, and not directly with Amazon. So if you want to integrate BookLikes with your Kindle account you'll also have to add Twitter as well.
In addition to letting members share their Kindle reading stats, BookLikes also supports Facebook and Goodreads integration. And if you have an account at LibraryThing or Lovely Books you can also import your bookshelf.
Book lovers use BookLikes to share their reading milestones, connect with book bloggers and share book reviews and book recommendations with their followers. The site offers a unique blog-like focus that lets each member decide whether they want to share a status update, post a snippet of text, or write an entire blog post.
The site is a couple years old, but its current focus dates to a May 2013 redesign that changed Booklikes layout as well as how members interacted.