The princess who travels with Zidane in Final Fantasy IX is Garnet.
Garnet is the princess of Alexandria and the daughter of Queen Brahne. At the beginning of the game, Zidane and the Tantalus theater troupe intend to kidnap her, but Garnet secretly wants to leave the castle after becoming alarmed by her mother’s behavior. She joins Zidane and the others, turning an attempted abduction into the start of her own journey.
Garnet later travels under the alias Dagger, a name she chooses after seeing Zidane’s dagger. Her story involves political responsibility, the discovery of her true origins, the destructive use of eidolons, and her struggle with grief after Alexandria is attacked. She is both a major party member and a summoner, using restorative magic and powerful eidolons in battle. Yuna belongs to Final Fantasy X, Terra to Final Fantasy VI, and Rinoa to Final Fantasy VIII, so none is the princess accompanying Zidane.