I don't understand what the joke here is supposed to be.
The other mecha's AI represents the straight-laced female character that gets introduced later, usually a class president (which is why she's called 'prez') that usually falls for the male protagonist, and is destined to not be the one otaku-kun picks as his girlfriend
GAL was upset that otaku-kun was 'seeing another girl' while she was gone, so to mess with both of them she force ejects him to make it seem like 'prez' wasn't helpful with his pilot training
Although it doesn't have an 'otaku-kun' as the main character (it's a teacher instead) the one manga I can think of with this situation is Gal Gohan