It turned out the exact combination Toya received from his parents' ice and fire Quirks, was self-jeopardizing. The doctor stated Enji designing his children through Quirk marriages was ill-advised. Though when Toya's Quirk began burning him, Enji finally saw a doctor over the issue. Enji wanted to press on with Toya's training, dismissing his unexplained hair change. He'd even call his father by his hero name on occasion. His father's legacy as the Number 2 weighted heavily on Toya, feeling more pressured to be a hero then his school peers. To enforce this, Toya was given fire-themed items like shirts. He put his dreams onto Toya, promising he could become Number One and surpass All Might. Likewise, Toya was very excited to learn all of his father's moves, including his strongest ones. In the beginning, unlike Shoto's relationship with his father, Toya's relationship between his father was actually relatively healthy Toya's training didn't have the level of harshness as Shoto's and Enji spoke to his eldest son more casually with the latter eager to learn new moves.
It wasn't exactly what he was looking for, but he saw its potential, so he trained Toya from an early age to be a hero. When Toya was born with a stronger fire Quirk than his father's, Enji wavered between disappointed and satisfied. Toya was conceived in a quirk marriage by Enji, to breed a hero heir that could surpass All Might. In fact, Enji's overwhelming negligence on him seems to be the main reason for his fall to villainy.
Dabi holds a huge grudge against Enji for rejecting him in the past and feeling like he was being used as a tool. The Flame Hero: Endeavor is Dabi's father.