I loved these moments. Not once did he correct her when she called him 'Monster'. The way I see it, they both feel like monsters in their own lives. Knowing how much it hurts to be the monster, and to be alone with that feeling, he couldn't bring himself to tell her his name and make her feel as alone as he feels.
If anyone has ever read the stories about Sherlock by Arthur Conan Doyle you would know that Sherlock has achieved greatness. He may have been born clever, but he admits that he trained his brain to deduce like it does.