Bruce has a habit of accidentally adopting superhero kids who aren't his own.
Roy comes over to see Jason one night when Ollie is being a shit, and he ends up staying for breakfast because "Well, Alfred's cooking, I can't refuse!" He lets Bruce give him a hug before he leaves, and will forever claim that he wasn't crying, no matter what Jason says.
When Clark is being an idiot about Kon, Tim brings him over to stay at the manor for a bit. For the first few days he's there, Bruce hover all. the. time. Finally Tim has a very firm talk with him about boundaries and personal space, but he still makes Kon know he's welcome in his weird batty ways. Even after Kon and Clark eventually start working on their relationship, he still calls Bruce dad.
When Damian first brings Jon over to hang out, Bruce just assumes he's his at first - black hair, blue eyes, he looks just like the rest of Bruce's kids. He doesn't even realize until the next League meeting, when Clark shoots him a look and says "Really, Bats, you didn't have to steal both of my kids."
At some point, after a particularly hard mission, the Manor is the closest safe house, so Bruce comes home to a group of teen titans passed out in his living room. They all wake up covered in their own blanket, matching the color of their costume.
As soon as Bruce finds out that Captain Marvel is really Billy Batson, he all but adopts him. He looks after him at league functions, and makes sure to be there for him if he needs to talk. He even offers him a room in the manor, but Billy declines, much to all the batkids relief. (They love Billy, they really do, but the house is getting crowded, Bruce, please go to therapy.)
Eventually, all the superhero mentors are terrified to mess up, because rumor has it if you do Batman will steal your kid.
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