What is M's real name?

"M" could be the British short form for Mom.
Mom is the British word of respect for a woman.

James Bond refers to her using both M and Mom in the film.

British military may refer to a female officer as Mom instead of Sir.

It's disrespectful to refer to a man as Mom, which is why Bond might have referred to her replacement at the end of the film as M. He was the guy who got shot in the arm (sorry, can't remember his name).

I remember when Bond calls him M he says it almost sarcastically.