Ppppprobably shouldn't go in the gameplay discussion, but since you asked, imo it's about player projection contrasting with the character's personality.
Robin in Awakening is a pretty bland character. Better than Kris in FE12, for sure, but is generally just a smart, nice, and levelheaded dude who acts as a straight man to all the zany characters that surround him/her. While he has a vastly increased role in the story, his/her background isn't that different from FE7's Mark, the original player insert of the series. He's the tactician - he/she does what the player does and reacts the way a reasonable person would in most situations. As such, it's much easier for the overwhelmingly male playerbase to project their own personality onto Robin. The character's design is pretty good for both genders too. F!Robin is more popular in Smash, but I'm willing to bet that in Awakening proper, the numbers were more even/skewed towards male.
Corrin, on the other hand, is more of a character than any previous avatar. He/she is the central character of Fates and drives the plot in a big way. They couldn't make him/her as simple as Robin because he's the protagonist, not a supporting character. But his/her personality is pretty divisive: he/she is naive, gullible, and childlike in a lot of ways. These traits don't correspond well to the cultural standards of a Western male hero. (Note: I haven't finished Conquest, but in Birthright, he/she stays static and makes it by on the power of heart for pretty much the whole game.) On the other hand, the traits fit common anime girl stereotypes better. A sheltered princess goes out into the world, optimistic and naive, and reveals strength in the face of adversity ("Silk hiding steel," if you will)? It sounds more natural to the regular consumer of these things. (2nd Note: Not gonna cast judgement as to whether that's a good thing or not.) M!Corrin thus fails as a player avatar and doesn't live up to cultural expectations as a character, but F!Corrin doesn't have to succeed as an avatar for most and is more palatable as her own character. In terms of design, M!Corrin also gets shafted on the customization screen and looks dumb in the steel action unitard (TM) that they both wear. F!Corrin gets a much better range of options there.
My two cents on this issue. Not gonna hate on players who use M!Corrin though, even if their character is a total dork
...Why is this my longest post on smashboards