In one of the opening cutscenes when starting the game (I think it was the Gridania one) your character sees a moogle and the moogle is surprised that you could see him. Apparently most people can't.

I'm not quite sure how the lore explanation fits that together with the existence of the Moogle Delivery Service. Perhaps whatever explanation is used for that could be extended to player characters. (E.g. Perhaps moogles can choose to be visible or not, but people with the Echo can see them regardless. Then player characters could be moogles who are choosing to be visible to everyone, just as the delivery moogles apparently are.)

It would require a bit of race-specific story exposition to explain them, and currently there isn't much in the game that's race-specific, but I don't think that would be insurmountable.