How do people visually recognize their own reflection?
The brain does map different parts of the visual image to different neurons. It's called topography and is a fundamental feature of all sensory neurons. 

The next part of your bolded questions jumps upstream. Some fMRI studies found "Jennifer Anniston" cells- cells that only respond to the image of Jennifon Anniston. Italian researchers found neurons in awake primates that increased firing in response to another monkey doing something. The researchers called these mirror neurons. 

So, there may be single neurons in humans that bind visual perception and recognition. However, it is more likely, that interacting groups of neurons change their activity relative to each other or the omnipresent local field oscillations. This mechanism appears in other places in the brain and, theoretically, it is also more robust to injuries.