Yuji Horii co-owns the copyright to the Dragon Quest franchise.
Dragon Quest,[a] previously published as Dragon Warrior in North America until 2005, is a franchise of Japanese role-playing video games created by Armor Project (Yuji Horii), Bird Studio (Akira Toriyama) and Sugiyama Kobo (Koichi Sugiyama) to its publisher Enix , with all of the involved parties co-owning the copyright of the series since then.
This is not well known, but you're right. Square Enix do not own Dragon Quest. Armor Project and Bird Studio co-owns the franchise. Horii is literally the one in charge and everything goes through him. He's the one who makes the decisions of who develops the games and the direction the games take. Fun fact, nearly all Dragon Quest games weren't made by Enix or Square Enix.
Square don't own Kingdom Hearts either, so 2/3 of their biggest franchises they don't even own.