Okay, I'm a Navy veteran with four years' active duty under my belt, and I swear I had no idea why the characters had the various ranks they had, which became especially annoying when they started bickering over them. I'm guessing prior service (as in WWI) played a major part, as about all the characters were old enough to have served back then? But that's just a guess...
shareAlthough the movie does not explain why the characters got the ranks they had, it seems it was partly based on the ranks of the real people the characters were based on, and partly just made up.
All the real life counterparts of the characters were already on active duty when they became members of the MFAA, and some had been reserve officers for many years before the war. George Clooney's character was based on George Stout, who was a Navy Reserve lieutenant. The counterpart of Matt Damon's character was a lieutenant in the Army. The Bob Baliban character was based on Lincoln Kirsten, who really was a private. Apparently the Army would not make nim an officer because of his political opinions, and maybe for other reasons as well.
The Bill Murray and John Goodman characters were loosely based on people who were officers in real life. It could be that since they are fictional characters, even though loosely based on real people, and since some MFAA members were, in fact, sergeants, the writers decided to make them sergeants.