Is it possible to rename a key in the Firebase Realtime Database?

I think I see what you're trying to do. Firebase doesn't have the concept of "renaming" a part of the path via update. Instead you will have to completely remove the existing node and recreate it. You can do that like so:

var booksRef = firebase.database().ref('books');
booksRef.child(oldTitle).once('value').then(function(snap) {
  var data = snap.val();
  data.bookInfo.bookTitle = newTitle;
  var update = {};
  update[oldTitle] = null;
  update[newTitle] = data;
  return booksRef.update(update);
});

This will remove the info from books/oldTitle and re-populate it with a new title in books/newTitle.

Caveat: This relies on reading the data and then performing a second async update. If you are likely to have multiple users operating on the same data at the same time this could cause issues. You could use a transaction to do this atomically but if /books is a top-level resource with many nodes that may cause performance problems.

If one person is likely to edit the data at a time, the above solution is fine. If not, you may want to consider using a non-user-controlled identifier such as a push id.

I would still use push(), that's really the proper way to store multiple child objects under category-node in firebase I think. For searching, have you thought about doing this on the client side? For example you can save all your book objects under books node, and retrieve those to search locally when you need to.

Also as ZenPylon mentioned, push() gives you a unique id that you can also assign as a prop to each book object so each book has a reference to where it is stored in the database.

You can export all the database as JSON, rename it in your favorite editor import it again to firebase via import/export option on the top right in the console window

One thing worth noting is that you can grab the unique key from the push() function.

var booksRef = firebaseRef.database().ref('books');
var newBookRef = booksRef.push(myBookData);
var bookKey = newBookRef.key;

Source: Firebase Docs - push()

I can't tell from your screenshot where the second instance of bookTitle is (is it the root element in the screenshot?), but if you don't want to use push(), you could grab the data at that location, remove(), and then call set() again, this time with the updated book title. Something like:

var bookRef = firebaseRef.database().ref('path/to/books/book1');

bookRef.on('value', function(snapshot) {
    var bookData = snapshot.val();
    var newData = {};
    var newTitle = 'NewTitle';

    bookData.bookInfo.bookTitle = newTitle;
    newData[newTitle] = bookData;
    firebaseRef.database().ref('path/to/books/' + newTitle).set(newData);

});