Bruce or rather Batman is adopted by all of the kid heroes as their father/ a father figure. It started when Dick (as Robin) called him in to help train the Titans (he also just wanted his papi to come visit) when he was leading them. They were hesitant at first but eventually trusted him partly because they trusted Robin and by extention who trained him, partly because it was Batman.
As time went on, it eventually spread to Young Justice, Birds of Prey, even the Outlaws, etc. also hearing about it as the heroes joined or siblings worked in different teams and called on Batman and he came with solutions and answers and comfort, albeit awkward.
The thing about Batman is that he answers the questions, comes when they need him, listens to them ramble, patches up their hurts and stays when they ask. He's there when they need him so they gravitate to him even when they don't.
The league doesn't notice... they never seem to when it comes to their charges and mentees.
Don't get me wrong, the original lineup absolutely adore their brother, but technically only the Flash or rather Barry knows (mainly because his protégés actually speak to him about things and because Barry kinda adopted Bruce too, though to him Bruce is more of a big brother than dad but kind of a dad nonetheless). The remaining original leaguers adopted Bruce as their little brother, except for Hal cause they act like the same age.
It doesn't mean they're any less protective of him, both in general and in battle, he is the only actual baseline human on the original team after all. The original leaguers are overprotective but know better than to cross his boundaries. They respect him and his expertise, for all they want to protect him.
Bruce never notices or rather realises. He loves them and supports them, but that's just how he is. He doesn't know they notice or that they care so much. They want to ask him for what his to them, but they're waiting (stalling) for the right moment. Obviously he knows that vaguely he is a father/ mentor type to the Gotham vigilantes. They call him dad (and equivalents they're comfortable with using) for the love of god.
But Bruce is unaware that his kids or any kids or even any hero for that matter, love and adore him. Like ye They idolise, admire, revere him...they're hero-worship for Batman is he only that never wore off and the only one they'd never admit to. The truth? They just say that to him because who would ever admit to Batman (their DAD) that he's their idol. Who are they kidding he probably knows anyways. He doesn't but the kids don't realise. It's not like they could ever bring themsleves to deny it reliably. They always get that look on their faces when talking about him. No matter what they actually say to him.
Even his villains are super protective of him and kinda in love him (most platonically, but some are romantically even though they know they don't have a chance with him).
That's the thing about Batman, about Bruce Wayne, whomever meets him, the real him, falls in love with him and would scorch the earth for him, be they a hero, a villain or any other.
He also misses all the signs of the heroes' adoration and admiration. All the heroes, even the kids, including the ones he raised, think he knows how much they love him. The truth? He doesn't see or know shit.
Bruce also knows all the kids' names, partly because they're friends with his kids and partly because he's helped train them from the beginning.
Bruce's canonical kids call him dad and stuff openly and show their want for his approval as openly as the others because they're just as eager and excited, but because they can get his attention in Gotham, they are kind enough to take a step back so the others may receive it, even though they're are pretty possessive.
