In a universe where Damian was raised by Bruce since birth and never was told his siblings were adopted
Damian, age 12 helping unpacking things at Dicks new apartment: whos that?
Dick: Who?
Damian: That guy in the photo with you
Dick:
Dick: you mean- my dad? I never showed my parents to you?
Damian: there is no way thats our dad
Dick: our? Damian you- you know I’m adopted right?
Damian:
Dick:
Dick: we are all adopted, Damian.
Damian:
Damian: Even Cass?
Dick: you were there at her adoption.
Damian: I’m twelve! Probably didn’t payed attention to it.
Dick: you never asked yourself why do we don’t call Bruce ‘dad’?
Damian: I thought it was some petty teenager thing!
Dick: I’M 26!
Damian:
Damian: is Tim adopted?
Dick: yes?
Damian: knew it.
In an universe where Damian was not raised by Bruce since birth (I forgot), but joined the family after Tim, and never was told his siblings were adopted (I focused on this and ran with it).
It already starts when they first meet. Damian introduces himself to Bruce as his only blood son and tells Bruce that he needs to get rid of the others because he's destined to be Robin. Bruce is surprised. He makes sure to tell Damian that everyone here is family, that the other children are his children as well and that Damian's mother lied to him. Damian later looks at the other kids and sees how they resemble his father. He shakes his head and concludes that a part of the Brucie is a playboy rumors are true apparently. His father just doesn't want to tell him, because he's too young to know about how kids are made or Bruce is too ashamed of his reckless actions.
Shortly after Damian joins the family, Bruce asks Damian to come to his study to talk about his birth certificate, custody papers, and a related story:
With Bruce and Talia not sharing a last name, Bruce decides Damian gets to choose his last name. Damian chooses Al Ghul-Wayne. Richard is called Grayson-Wayne, so Damian can have his mother's and father's name both as well.
They also discuss the story that Bruce is going to tell the world about how Damian came to be. It's a fabricated story that focuses more on Brucie's playboy persona and how glad he is to meet Damian after he thought he died during the pregnancy than on who Damian's mom exactly is and why Damian is so good at turning everything into a weapon at school. Bruce and Jason also had to come up with a story to tell the public about how Jason left Bruce and went to his mom, got hurt badly (he died and got ressurected) and lost his memory, and came back years later.
Bruce writes his name down under the header "father" on the birth certificate and custody papers. Making himself officially Damian's dad and a part of the Wayne family and manor. The papers get send out to the official place where they should go and a copy gets put in a locker where the other children's papers (adoption papers) are also put. Damian is very happy.
He's also happy when Cass' papers get added to the locker, but he doesn't realize that his papers are different from the other papers (adoption papers). Not realizing how Bruce not referring to Cass' parents by parental title but by Cain and Lady Shiva and how Bruce telling Cass that he wished that he met her sooner so he could've take her in before the painful training does not mean that Bruce could've come earlier and taken custody of her and taken her to the manor and discuss birth certificates and be a happy family together with him as her dad (like Damian sometimes wishes Bruce could've done for him).
They hold an after-party with "welcome to the family!" and "congrats, it's a girl!" banners. Similar to what they held for Damian when he joined the family. His family is just extra and late with parties like that.