Zodiac System, expect it depends on what Summon that character can summon. Aries = Ifrit, if you give Ashe the Zodiac " Aries " she'll summon Ifrit throughout the whole game. You cannot change or remove though.
Off topic, because this one is my favorite FF, but I really like this idea. All the Espers are connected to a sign anyway, so why not do this in the new one?