AU where Severus expects a mini spoiled bully James to show up for sorting but instead he sees Harry as he is - unfed to the point of starvation, arthritis from being locked in the cupboard for his whole life, painfully starved of affection and any positive attention and Dumbledore goes "hmm how unfortunate. but he has to stay because of the BLOOD WARDS-"
So Severus stays quiet and then fucking kidnaps him right off the platform at the end of year 1 and goes on the run. He tries to be nicer for Harry's sake, but he's never been a nice/warm person. Hes too socially awkward - didn't get the chance to build those skills at Hogwarts. Either way Harry goes with it because Severus is actually feeding him as much food as he could want and letting him breath fresh air and he's got 200% less chores so it's already better than the Dursleys.
The kidnapping makes front page news. That front page makes its way all the way to Azkaban and Sirius Black sees it, loses him shit in worry (he's got no idea that Severus was a spy - he thinks Severus is taking revenge for Voldemort and all the bullying James did). Sirius breaks out and hunts Severus down.
After a very messy encounter where Sirius tries to kill Severus for thinking he's hurting Harry, Harry tries to kill Sirius for betraying his parents, it eventually comes out that Pettigrew was the traitor. Sirius thinks that Pettigrew, the rat, died in the blast. Severus mentions that only 1 of Pettigrew's fingers was recovered. Harry mentions that Ron had a rat that was missing one finger. All of a sudden they're all on the same side.
In the quest to capture + reveal Pettigrew, Harry bonds with Sirius who's 1000% more emotionally avaliable than Snape - and actually has stories about Lily and James as parents. But neither of them can leave Snape because he's the only one with a wand that's not covered with the Trace; and they need magic to evade the ministry and Dumbledore.
Pettigrew, seeing Severus going missing and Sirius coming for him, goes to raise Voldemort a few books early. Harry starts having horcrux dreams. He tells Sirius and Snape. Unlike Dumbledore, they don't sit on their asses - they study the scar, find out it's a horcrux and get it out of Harry. They realize it's one of many and hunt down + destroy the rest and clear their names.
Harry grows up with two father figures - one stern, but quietly caring, one very affectionate and loving, but quietly broken. Sirius is the parent Harry goes to when he's worried about stuff like girls, friend dynamics, gof/people trying to kill him. Severus is the parent Harry goes to when someone like Umbridge abuses him - Harry knows Sirius's reaction will be Loudly Protective and he can't handle that - Severus's reaction is Quiet but no less protective.
They're both strict as hell and the first time they catch Harry sneaking out with the invisibility cloak Harry gets into so much trouble he almost sends Hedwig to Hermione or Ron for help. Sirius is the one that lectures. Severus decides the punishment. Both of them assure Harry that it won't include his basic right being taken away (like at the Dursleys).