This may seem somewhat implausible to some, but I feel like when Sirius got the story of why Weasley Kid #3 wasn't coming to Grimmauld Place in OoTP, the story probably sounded a bit familiar to him.
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Sirius Black has never met Percy Weasley.
Well, that's not quite the truth. In the year Sirius spent wandering the Hogwarts as a dog, he saw him often enough. He would see a tall, red-haired seventh-year Prefect running around the school grounds trying to restore order and help out his classmates. He was always having to get younger students or their pets down out of trees or break up fights or take classmates to the hospital wing or to their Head of House for punishment. This he would do for Gryffindors, for Hufflepuffs and Ravenclaws, even for Slytherins. The boy was the type who was scrupulously, obnoxiously fair; Sirius was irresistibly reminded of another red-haired Prefect he had known.
On one occasion, the boy had seen him and approached cautiously, his wand out. His Head Boy badge was highly polished, and if a dog could grin, Sirius would have; James had always gone around with his own Head Boy badge so polished it gleamed like a lantern in the sun. Then, the boy had gone down on one knee and held out his right hand, passing his wand to his left. "Shake if you're friendly."
Sirius had been so amused by this seemingly uptight boy's sudden show of humor that he'd shaken his hand.
So Sirius has met Percy Weasley. He's even shaken his hand, though Sirius can guess that Percy would be horrified to learn that he'd shaken the hand (or paw) of a convicted murderer. But Sirius has never really spoken with Percy Weasley. He's never spoken with him, and doesn't have Percy's side of his own story or anyone else's. That's become important, now that Sirius has become familiar with the rest of his family.
The Weasleys have come to take up residence at 12 Grimmauld Place and joined the Order of the Phoenix, sans one. Percy has not. Percy has "sided with the Ministry." Percy has "turned his back on his family." As far as many of the Weasleys, chief among them his younger siblings are concerned, they are "well shut of him" and they "always knew he'd do this, the great prat."
Sirius finds himself in front of the Black family tapestry, taking a swig out of a whiskey bottle and thinking about the past. About himself, and the circumstances that led to his name being blasted off of this tapestry.
There was a boy once, who was born to a family that he didn't quite fit into. At first, it wasn't so bad, but as time wore on and he grew more and more obviously different from them, and grew less concerned with toeing the party line, and his family closed ranks, and made it increasingly unbearable for him to live under the same roof as them. Eventually, there came the last straw, and the boy left.
Is he thinking about Percy Weasley, or about himself?
Sirius isn't sure, anymore.
He's resolved to stay out of the situation as best he can; this is something the Weasley family needs to handle themselves, and frankly, the more Sirius finds out, the more uncharacteristically uncomfortable he feels about the whole thing.
Because as far as he can tell, Weasley family politics are not that different from Black family politics. His parents, his cousins, and to a lesser extent his brother Regulus (who seemed significantly more ambivalent about the whole thing) began to exclude and ostracize him, first when he was sorted into Gryffindor, and later when it became clear that he was not going to go along with their pureblood supremacy ideology. As Sirius recalls, Ron not all that long ago, before Percy cut ties with his family, said something to the effect that he wouldn't put it past his brother to hand them all over to the Dementors if he thought they had committed a crime. If that is what his other family think of him, and that is how vocal they've been about it, something tells Sirius that Percy knew quite well what the rest of his family thought of him long before he decided to leave home.
In all honesty, Sirius would rather not be drawing these parallels. He likes the Weasley family. He likes Arthur and Molly, no matter how often he butts heads with the latter. He likes the Twins, loves their sense of humor and their eagerness to exercise it. He likes Ron, glad that Harry has such a loyal, reliable friend. Sirius likes the rest of them as well, and he'd rather not compare them in any way, shape or form to his own family. But there's really no denying it. Weasley family politics are not that different from Black family politics: you toe the party line, or you get kicked out. The Blacks blast you off the family tapestry and refuse to ever speak of you again in polite company. The Weasleys declare your name mud, and refuse to ever speak well of you again in polite company. Apparently insulting you constantly and assuming the absolute worst of you is completely fine; Molly's certainly the only one who seems comfortable being seen to think well of her third son.
Though Sirius barely knows Percy Weasley, has never spoken with him, he can't help but sympathize with him, for all that the boy is being wrong-headed, and for all that he does not at all appreciate anyone implying that Harry is wrong or lying about something he quite clearly saw happen. The boy was told that he was wrong to be proud that he had been given a high-ranking post. He was told that he was wrong, that he was being an idiot, that he'd just been given the job to spy on his family, and that if he did not fall in line, all familial support would be withdrawn and he could consider himself unwelcome in his own house (Or perhaps just more unwelcome than he was probably already feeling).
But Percy did not fall in line, and he did not cave; he stuck to what he believed, refused to believe that he'd been given his post just to spy on his family, and perhaps just to prove that he struck out on his own. Ron and the Twins often talked about how ambitious Percy was, with perhaps the implication that they now thought that he would have been better off in Slytherin House. Sirius couldn't agree. Wrong-headed it might have been, but damn if it wasn't a Gryffindor-ish thing to do. Nothing short of courageous, though it doesn't seem to be the sort of courage the Weasley family particularly values, and it isn't the sort of courage that Sirius himself learned to value until recently.
Sirius sighs, running his fingers over the black, still-sooty spot where his name once was on the Black family tapestry. Inevitably, his eyes fall to the intact spot just next to it.
Regulus Arcturus Black. 1961-1979.
Sirius doesn't know exactly how his brother died. He knows Reggie had fallen in the Death Eaters—he probably has dear cousin Bellatrix to thank for recruiting him. Sirius likes to imagine that Reggie had eventually realized that what he was doing wasn't a game, but that by the time he did, he was just in too deep, and it was his attempts to escape that had brought about his death. But he can't know. No one knows exactly how Reggie died except the one who killed him, and whoever that is, they aren't talking.
He sees himself in Percy, but Sirius starts to see Reggie in Percy too. The more time goes by, the less convinced Sirius is that Reggie was just another fanatic, cut from the same cloth as Bellatrix, and the more he thinks that Reggie was just a stupid kid who had listened entirely too much to Mum and had bitten off way more than he could chew, and choked on it. Percy seems much the same, except he hasn't choked, at least not yet.
The day Sirius found out Reggie was dead was the only day Sirius ever regretted leaving home. For all their differences, he'd preferred Reggie alive to dead, and he knew that if he'd been there, if he'd made an effort, he might have been able to influence Reggie not to join up with the Death Eaters, or at least to get out soon enough to save himself. No such luck. Reggie had found himself completely alone in a den of wolves, and ended up devoured. At present, Percy is alone in a den of wolves, because as much as his family claim he turned his back on them, they've turned their backs on him, because their wayward son didn't toe the line. Just like Reggie and Sirius.
Sirius just hopes, really hopes, that Percy has better luck than he does, or than Reggie did.