Damian Wayne revealed!

Inner Demon

Jason had killed, and he saw the logic in it. He wasn't as soft as his other 'brothers' and he knew how the League did things...
He understood.

Hence why Damian considered him as an 'ally' of sorts, after his demon that is. He also went to him for help/ advice when he didn't understand something the others did. (Though the others knew this; at least that Damian sneaked of towards Crime Ally sometimes and they would maybe find him wearing Red Hood's jacket or smell a whiff of cigarette smoke.)

As for the rest of his paternal family...

His father's parents were dead so he held no interest in them,
said father also refuses to acknowledge his training and him.

...Nightwing, otherwise known as Richard 'Dick' Grayson, was.. strange but not necessarily bad, annoying and confusing (quite often), but somehow he reminded him of the demon who helped him after a brutal training session at the League. So he accepted/ tolerated his presence near him. He cared.

Jason Todd was Red Hood, though he had turned from being a Crime Lord to being an Anti-hero that semi-worked with them and used rubber bullets; he was the only one that killed when he deemed it necessary. He understood.

Timothy Drake aka Red Robin was... well, he didn't like Damian and neither did Damian like him; another reason on why he rather associated with Jason if he had to choose who to agree with during a foolish squabble between them. Unless Jason insulted his mother, then he was on no-one's side.

Both he and Jason had tried to kill Timothy and used the nickname 'replacement' for him, which fit him perfectly by the way. He could never hope to compare with the original, the bloodson of Batman.
But Timothy was smart, smarter than the first two Robin's and having gained the title of 'Worlds 2nd Greatest Detective' from his father who was ranked #1. He had figured out his father's identity and made him take him in, becoming the next Robin by himself - as a child.

So he had semi-respect for him to come up with a plan that actually worked and not get himself killed while at the same time being acknowledged by his father. Something he desired.
He rivaled him.

The only butler of the house, 'a close family friend' of his father that was also the only staff in the mansion, Alfred Pennyworth, was acceptable. He took impeccable care of the mansion and could convince his father to stop whatever he was doing simply by saying 'Master Bruce', same procedure with the others to the point he had to acknowledge Pennyworth.

The elderly man also didn't necessarily adhere to the no-killing rule, adding to his ninja-like skills.

He sometimes wondered if the man could actually read minds or see the future due to his behavior and preparation. Out of respect and... attachment for the butler, Damian had named the first cat he got after him - the man had seemed happy if not honored about it. He supported him.

Another thing Damian grew attached to in the course of his stay in Gotham: the animal's.
He had been allowed to keep pets, something he wouldn't have been able to do at the League as only weapons were necessary.
Not to mention more than one and not just the regular kind.
Keeping Goliath added on to that.

.
.
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And when he saw how big his family had grown and expanded during his stay he felt... pleased. He'd even gained a sister, well more than one, as they had worked with the previous Robin's and even had relationships with some of them. Meaning they were all older than him, making him the youngest among them. Alfred the butler clearly being the oldest among the ragtag family of bats, birds

and cats.

Along with a sister was another mother figure, who had reformed from cat burglary to something similar to Jason. Even when he was rude to her she took it with humor, like Grayson. Eventually he accepted them both as a kind of other parent-figure. Jason often referred to it as 'mother-henning'.