Harry would likely be more confident. Sirius is a bold and brave character, and growing up with him would instill those qualities in Harry more strongly. He wouldn't be as meek as he was in the early days at Hogwarts.
One possible plot twist could be that Sirius Black, being a wanted man, has to constantly move with Harry to avoid capture. This would make Harry's childhood very different from the one at the Dursleys. He'd grow up learning about the wizarding world on the run, perhaps meeting some unique characters in hiding places.
Fanfiction about these three can take many forms. For example, there could be time - travel fics where Harry goes back in time and meets Regulus and Sirius when they were younger. This could lead to all sorts of interesting plot twists, like Harry trying to prevent Regulus from joining the Death Eaters in the first place. Or it could be a fic where Sirius survives and he and Harry explore the secrets of the Black family together, including things that Regulus left behind. There are also fics that focus on the emotional bonds between the characters, showing how the different personalities of Regulus, Sirius, and Harry interact and influence each other.
In this kind of fanfiction, Harry would likely grow up with a more hands - on and perhaps wilder form of parenting. Sirius was brave and impulsive. He might teach Harry some of his more unorthodox magic skills earlier. Harry would also be surrounded by Sirius' loyalty to his friends and family, which could shape Harry's own values in a different way compared to growing up with the Dursleys.
His relationship with Sirius would be extremely strained. Since Walburga was Sirius' mother and they had a very difficult relationship, Sirius would likely see Harry as a product of his mother's influence. Harry, on the other hand, might be initially hostile towards Sirius due to being raised with Walburga's views.
The story of Fem Harry Potter raised by Sirius Black could be full of twists. Sirius would have to hide her from the Ministry and Voldemort's followers. This would mean a life on the run at times. But it also means she'd be more street - smart in the wizarding world. She might grow up with a love for Sirius' motorcycle and a sense of adventure. Her time at Hogwarts would be different too. She'd have Sirius' stories in her head and might question the school's rules more often, leading to some interesting situations with the teachers.
One possible reason could be a misunderstanding. Maybe some false information was spread about Harry that made Sirius believe he had done something unforgivable. For example, if there were rumors that Harry was involved with dark magic in some way, Sirius, who has a strong aversion to the dark side, might be quick to disown him.
One possible plot could be that Harry grows up with a stronger sense of family. Sirius and Remus would teach him about his parents more directly. He might be more confident in his magic earlier, as they could start his training sooner. Also, he'd likely have a different view of werewolves and Animagi from an early age.
One possibility is that Sirius would teach Harry all about his family's history in a more private and laid - back setting in France. He might introduce Harry to different magical cultures unique to France, like the Beauxbatons Academy. Harry could grow up with a more carefree attitude, away from the prying eyes of the British wizarding world.