Both Catherine and Major Paul Petkoff are minor characters in Shaw's Arms and the Man, so their characters and roles are not highly developed by Shaw. Nonetheless, each plays an important role in the progress of the play. Catherine is the matron of the play, a mature woman who has a place in the social order of Serbia won by her husband's newly acquired wealth, the same wealth that earned him the military rank of Major. In this regard they are similar for having attained prominent positions through accumulated wealth.
Catherine is the symbolic representation of Shaw's theme of the fallacy of romanticized ideas of love while Major Paul is the symbolic representation of the fallacy of idealized notions of heroic warfare. In this regard they both are similar in that they each represent the negative themes that Shaw is presenting. By way of contrast, Major Paul's focus is only on heroic military idealism, while Catherine focuses on both romanticized love and heroic military idealism.
Catherine is important to the plot of the play because her presence and traits show clearly why Raina has the beliefs and attitudes she has, indicating an opinion that the ideologies Shaw objects to are passed on as part of a legacy of social wisdom. Major Paul assists the plot by being the one through whom social wisdom meets real wisdom. It is his robe that brings Bluntschli back to the house; it is his robe pocket that ignites the moment when truths are told and lives change course.
This means that indirectly, Major Paul is responsible for his wife agreeing to the genuine love and marriage between Bluntschli and Raina (on condition that there is proper wealth and prestige attached). So in this sense their characters contrast: Major Paul is a vehicle for sincere thought and feeling to enter their lives and decisions while Catherine only yields to sincere thought and feeling when she sees a material advantage to doing so.
Finally, in a sense, Major Paul is a foil to Bluntschli. A foil is a character that emphasizes the protagonist's good qualities by displaying the negative side of the same qualities and who usually has a major characteristic in common with the protagonist. In this case the qualities pointed out are Bluntschli's sincerity, genuineness and reasonble thought. The characteristic Major Paul and Bluntschli share is a military career.