Restricting the party members to specific classes serves to make them more unique in battle. More importantly, when it's tied into the story, it even bolsters the depth of their character. They did the same thing with FFVI.

Would Lydia's role in the game's story be nearly as special if every character could be a summoner? Would it be nearly as memorable that Lock demands to be called a treasure hunter if everyone could be a thief? Why not just let everyone turn into a crazy pink Esper while you're at it?