What is the significance of the character being referred to only as “the boss” in "The Fly"?

The visitor, Mr. Woodfield, and the narrator of the story both refer to the owner of the business as the "boss." This indicates how fully and deeply he is identified with his role in life. He has lived and breathed to be the boss.
But, as we find out in a way that twists our hearts, the title has, ironically, become meaningless to the boss. He built his business for one purpose: to leave it to his son. But his only son was killed in World War I. As the narrator tells us:

Ever since his birth the boss had worked at building up this business for him; it had no other meaning if it was not for the boy. Life itself had come to have no other meaning.

The boss is identified by a title that once gave his life all its meaning but now has lost all meaning to him. By calling him the boss, Mansfield emphasizes that he is nothing now but the empty shell of a man.
The impersonal title also suggests he is an Everyman, and that, given the heavy casualties in World War I, throughout England there are many "bosses" like him, living meaningless, anguished lives, feeling like the fly hopelessly drowned in ink while being told to keep their spirits up and to keep on trying.