I wonder if it might be Hemingway's way of comparing Jake & Cohn. In a way they are foils for each other. They both have feelings for Brett and both get hurt by her, but Jake takes it quietly and more maturely than Cohn who makes a big deal and gets overly emotional and upset. Perhaps Cohn represents everything that Jake is trying NOT to be- pushed around by women, a slave to his emotions and desires, overemotional, immature, self-pitying, a drunkard and a whiner. From the character's point of view, Jake could be talking so much about Cohn because he is subconsciously comparing himself to Cohn to make himself feel better. From Hemingway's point of view, I guess talking about Cohn shows that he is important and draws the readers attention to his purpose as a foil.
Of course, I'm not a professor or anything. I'm just another student slogging through an essay on a novel that I hate. Still I hope this helped!