I'm 99% sure it's all for show. If it did increase the rate of encountering a certain nature, I'd think I would of heard something across the multiple sources I check.
I have my nature on my card set to adamant. And lo and behold when I caught Reshiram and Tornadus they both had an Adamant nature.
I have to do more research and catch more legendaries but for now I think it is purely luck.