I can see what you mean about the crew having very different backgrounds and characters, but there are no doubts about their affiliations, except for Natalie, Sara, and maybe some of Minerva's illusions; every main character is undoubtedly part of the League's military.
Also, it will never properly happen that you will see all of them wearing a uniform at the same time, in contrast to the picture above. Illabs need special uniforms, Veludran nobles have extra privileges, Minerva officially doesn't exist, etc...
I'm glad you like the suits on their own.