I think “cat in the hat” and “cat in hat” should be close enough to count as the book. Even “cat” or “hat” alone if there are no other candidates for those words in scope, but, in the presence of an actual cat or an actual hat, “cat” or “hat” should be taken to mean the actual thing without need of asking for disambiguation.
So let’s make it more pathological! There’s not just a cat, but a cat curled up asleep in a ushanka! And, its name is Book! 
This discussion suggests to me a useful development aid. For each word the game has defined as possibly referring to something the player might mention in a command, if that word could match multiple things, output the word and those things. That is, offer the author some guidance to where they might want to tune disambiguation.