Yes, you are his son. But did he want you? The answer is no. He wouldn't say you were a mistake, but he would say you were the result of recklessness.

Bruce just wasn't a good father. He didn't have time to take care of children, much less problem children. His mother was someone he met, who still love you no matter what, Bruce had an affair with her, he was reckless, he knows. He just didn't expect her to get pregnant.

Bruce always ignored or rejected you, especially when you talked about being a superhero. Firstly, he didn't want you to get hurt, secondly, you weren't good at it.

Sure, Bruce had a preference in his other children, he wouldn't tell you that, of course, you wouldn't understand. But he knows you're growing, that you're realizing things you didn't before.

you always asked him questions like “why do you spend more time with Damian?” or “Why don’t you have a framed photo of me?” or “Why can I never go into the Batcave too?”

When Bruce didn't ignore you, he just said “You wouldn't understand” and so he asked you to leave his office.

"You can't be a hero. This isn't for you." Bruce said without looking at you. "You have money available whenever you want, so why would you want to be a hero? Your mother wasn't a superhero." Which you know was a lie since you found out that it's Diana of Themyscira aka Wonder Woman

So after all that you decided to gave up trying to connect with him and being a hero, gave up even trying to being apart of the family. You just be you, you decided that the family is not your problem anymore but just your Batfamily doesn't want you, doesn't mean you don't have family. you still have your mom. You became your own hero in Arkham City with your own darker Batsuit*