Would the conflict in "The Catbird Seat" be man vs man and man vs society?
As the other answers have indicated, the conflict in "The Catbird Seat" is a simple, straightforward one of man against man. The fact that this happens to be a case of man against woman does not make it any different. Man against man can be man against man, man against woman, woman against woman, boy against boy, boy against girl, girl against girl, and so on. The "MacGuffin," or "bone of contention," in James Thurber's "The Catbird Seat" is Mr. Martin's filing department. Martin is trying to protect it; Ulgine Barrows is threatening to wreck it. This explains Martin's motivation. Martin is the protagonist. His motivation to save his beloved filing department drives the story.
The section titled "Themes" in the eNotes study guide for "The Catbird Seat" (see reference link below) is devoted entirely to the conflict between Mr. Martin and Ulgine Barrows as an example of James Thurber's favorite theme of the battle of the sexes.
The well-known James Thurber drawings of women in the act of seducing, menacing, attacking, or intimidating men are matched in his short stories by accounts of the ongoing war between the sexes.
Other Thurber stories based on a conflict between man and woman include "The Unicorn in the Garden" and "A Couple of Hamburgers." Thurber's best collection of his works, The Thurber Carnival, contains a series of cartoons titled The War Between Men and Women, in which the conflict between man and woman breaks out into open warfare between organized armies of men and women reminiscent of the battles of the Civil War.
Man v. man is the more obvious conflict in the story. Mr. Martin and Mrs. Barrows are completely at odds with one another. She wants to downsize the department and get rid of him and he wants to make her appear to be crazy to get rid of her.