"The story of Doctor Who is always the story of the companion, it’s always their story. It was Rose Tyler’s story, it’s Amy Pond’s story – the story of the time they knew the Doctor and how that began; how it developed and how it ended. It is their story. The Doctor’s the hero, but they’re the main character." ~Steven Moffat on Doctor Who
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.
It may be sad but it is also very sweet, because it means that it isn't just the Doctor who has their names written on his heart, but also the Tardis. Because I believe that the console room is the heart of the Tardis.