"M." is clearly Freud's pseudonym for Dr. Josef Breuer, the long-time mentor for whose wife Freud named his first daughter, Mathilde. Breuer was first author of Studies on Hysteria, which had just appeared (Breuer & Freud, 1895). Breuer's recounting there of his treatment a decade earlier of "Anna O. (Bertha Pappenheim) has served both as a model for psychoanalytic case-writing, and as a cautionary tale of the perils of (counter)transference.
"M"'s appearance, and the "quickly" remind Freud of an incident when he caused a severe toxic reaction with a supposedly harmless injection of suphonol, and called on Breuer for help. The patient's name was Mathilde.
Freud's dissaffection from Breuer during the later '90s is well-documented in the Fliess letters. In this dream Breuer not only makes a fool of himself as a diagnostician, but he is physically belittled by being bartlos, stripped of his patriarchal beard.
The initial "M" also alludes, as Freud's analysis indicates, to his elder half-brother Emmanuel, whose limp has been appropriated in Breuer's walk.