Yes the "X" IN fact did have to do with Gen X, that all came out back in the 90's, I'm among that generation so I know. They were trying to appeal to us and our generation. If they're not playing any music from Generation X's era then they don't deserve the "X".
If that's what it meant (I have no idea, I personally don't think it's got any more meaning than Y, Z, etc) in 2013, X is no different than Q, B, Z, Y, etc. It doesn't symbolize anything, it's just supposed to sound catchy, like any name. The average listener doesn't read in to it, nor do they care. Alternative stations using the "X" name have plenty of listeners born in the 80s and 90s, myself among them.