Duke’s path to heroism begins before he ever meets Batman. It starts when he meets Bruce Wayne. He meets Gotham’s prodigal son during the Zero Year event, when his parents bring the unconscious Wayne into their home. When they meet, Duke is already studying up and planning to take on the Riddler by himself. He’s all of 10 at this point, and there’s a man who has taken over the entire city, and Duke is completely unafraid. The consequences for failing the Riddler’s test are dire, but it doesn’t matter. Duke is willing, Duke is ready.
After this, we see the next major development of Duke’s personality. Batman, who Duke has looked up to since he was rescued in the wilds of the Zero Year while fishing, is gone. And so are his parents. Duke enters foster care and essentially becomes a problem child. He has gained a deep distrust of authority and a fierce independence. He bounces around foster homes, refusing to stay put and be used, because he has learned to rely on himself for his needs. He’s had it rough. He develops his manifesto of “Robin doesn’t need Batman”, which is a perfect summation of how he operates as a hero. Further still, Duke chose the vigilante life on his own. He recognizes it’s crazy and irresponsible and he can’t help it, it’s called to him.
At this point he's joined and even leads a division of the We Are Robin Movement, and we see him developed a bit further in relation to Bruce specifically. By this point he has also deduced the identities of 3 members of the Batfamily (Bruce, Dick, and Damian), and knows that Bruce is hiding from himself [post-endgame]. He confronts him on it, calls Bruce on his bs, and that’s another important facet to Duke’s personality. He doesn’t take shit. At all. Duke stands firm in his identity and doesn’t take others being untrue to themselves lightly. Duke is perceptive enough to call someone’s bluff.
Now, while Duke is fully prepared for crazy and being a hero, he is still unsure of what that means in relation to the Batfamily. Note that this isn’t the same as feeling out of his depth; Duke would be a vigilante regardless of the others. He already was one. But now that he’s in the gold, he has doubts as to what that actually means. He’s not like them, but only because he’s not in the dark. He literally operates in the light. He’s not hiding. Because he’s not afraid of who he is and what is in him.