It's background and fluff mostly, which is as it should be.
Not all characters who have a background that reasonably allows them to do whatever a certain feat does will necessarily have attended whatever organization the feat is restricted to.
Furthermore, outside of house-ruling, you don't need to be taught a feat, period, anyway. Whether from a war college or a mentor. You just level up and take it, with few exceptions.
However, things that should be noted: To become a member of the Legion of Ash, you need to have attended and graduated one of the war colleges. If you haven't, you're out of luck.
Another thing that should be noted is that, pretty much, the war colleges produce PC class characters. Which is a big deal; most other lands have Warriors, not Fighters. PC's are the heroes, the exceptions, the folk who stand above their fellow man even when they come from lowly, peasant backgrounds.
For most people, however, that just doesn't cut it. The war colleges grant the training most people need to be a Fighter rather than a Warrior.
So it's not just bragging rights; those saying they graduated from Clayborne or whatever is saying "Hey, I'm deadly." They're saying they're a PC class with probably at least four or so levels. A PC hearing someone graduated from a war college is tipped off that the person's dangerous. Whereas as a background item for a PC, it well explains their ungodly power (which most PC's will have by about level 6).