My guess is that the player character can't be counted like a normal individual, and thus our soul(s) are unique. I have the theory that it was our original Amaurotine self the one which summoned Hydaelyn (with a different method probably involving linking her to us, with no tempering like Zodiark did with the Ascians after being summoned) to counter Zodiark. We most probably abandoned the Convocation because we weren't okay with summoning Zodiark, and after the other 13 members kept sacrificing half the population to make him stronger, it makes sense that we tried to do something about it.

In other words, I think that being directly connected to Hydaelyn is what allows us to get out of the lifestream and reincarnate forever with our soul "intact", as long as she exists (which also happens with the 10 Ascians that were split, but in this case due to Zodiark's influence). As for the rest of the people, I guess that once they get to the lifestream they get actually "recycled" and used to create a new one.

However, my other theory revolves around the fact that the goal of the Ascians was to sacrifice the new life to bring their people back *from* Zodiark (in other words, to swap them). As I understand it, that means that most of the souls are actually independent, and only the few that got split during the Sundering (original ancients, AKA current Echo users) can come back from the lifestream. Perhaps due to their unique original god-like capabilities.