The story is two-fold, since summoners throughout history FF history were capable of summoning multiple either through trials or training, i.e Rydia.
The main exception was Tina, whom was basically an Esper herself. It's why we need a continuation of the story in order to even become a summoner siince currently, nothing is really driving us to allowing it to be a class or job, especially given that we're basically set to kill one of the staple Summons in the series. So unless it all leads to an epic build up of ending up with Carbuncle as our first summon again then we end up summoning a new set, it seems they're tying Primals (summons) a lot more heavily into the story this time around.