I don't exactly see what makes this scene above and beyond everything else. What are some examples of cinematography and Mise-en-scène in this scene that really enhance the meaning of this scene?
The meaning of this scene was mainly enhanced by the improvisation of Robert De Niro.
The scene where Travis Bickle is talking to himself in the mirror was completely ad-libbed by Robert De Niro. The screenplay details just said, "Travis looks in the mirror." Martin Scorsese claims that he got the inspiration for the scene from Marlon Brando mouthing words in front of a mirror in Reflections in a Golden Eye (1967).
Another reference is found here which links this dialogue to Bruce Springsteen
Robert De Niro improvised that whole paranoid monologue, including what would become the movie’s most famous line. (The film's screenwriter, Paul Schrader, later said, “It’s the best thing in the movie, and I didn’t write it.”) De Niro got the line from Bruce Springsteen, whom he’d seen perform in Greenwich Village just days earlier, at one in a series of concerts leading up to the release of Born to Run. When the audience called out his name, The Boss did a bit where he feigned humility and said, “You talkin’ to me?” Apparently it stuck in De Niro’s mind.
Now, to ans your question, more specifically,the cinematography and the props rather the Mise-en-scène, were not the major reasons why this scene became iconic. We are talking about 1976, where in motion pictures, improvisations this long and detailed were quite rare.
Martin Scorsese here himself explains how the shot went.I would rather request you to watch it. Its interesting! It's at 3:57