Sirius Black loved Harry Potter, not James Potter’s son
In the fifth film, there’s a moment where Harry fires off a spell and Sirius calls “nice one, James!” It’s an interesting character moment, but not something that’s seen in the books. And it does a disservice to Harry/Sirius’s relationship as well as Sirius himself to imagine that Sirius simply saw Harry as a James replacement.
It’s impossible to completely separate Sirius’s relationship with James from Sirius’s relationship with Harry, because without James, there’s nothing linking their lives together. But I would argue that Sirius did see Harry as his own person and did not see him as a replacement best friend.
Sirius is not a strong authority figure, but he is an authority figure. I would say he falls squarely in the “helpful uncle” category. He doesn’t treat Harry like a child, but he doesn’t treat him like an equal either. He sends him warning letters, (lightly) reprimands him, buys him the broomstick as payback for missed birthdays, etc.
Keep in mind Sirius knew Harry as a baby. If you change someone’s diapers, it takes a few decades to get to the point where you see them as a peer. Sirius wanted to take responsibility for Harry as a baby and that he’s fairly quick to ask 13-year-old Harry to live with him in PA. You can argue that he was looking for a James substitute even then, but as illustrated above, he did see Harry as someone for whom he was responsible for protecting/caring for.
In the book version of the Department of Mysteries battle, Sirius doesn’t treat Harry like a fellow Order fighter. He goes out of his way to protect Harry and repeatedly tells Harry to get his friends to get out.
[When Harry is facing Dolohov]
“Sirius had hurtled out of nowhere, rammed Dolohov with his shoulder and sent him flying out of the way.”
“Nice one!” shouted Sirius, forcing Harry’s head down as a pair of Stunning Spells flew towards them. “Now I want you to get out of-”
“Harry, take the prophecy, grab Neville and run!” Sirius yelled, dashing to meet Bellatrix.
Sirius doesn’t view Harry as a dueling partner. He sees him as a 15-year-old kid who, despite his above-average DADA competency, still needs to be protected. Sirius would never tell James to abandon an Order mission, because James is a competent adult and Sirius’s friend. Harry is Sirius’s responsibility.
And yes, part of Sirius’s love for Harry was probably wrapped up in his love for James and his feeling of obligation toward someone who he viewed as a brother. However, that doesn’t mean that he didn’t love Harry as an individual.
Sirius doesn’t treat Harry with the kind of removed affection that you treat the loved ones of people that you yourself love. He seems to genuinely care for Harry in his own right.
Sirius was pretty solid when it came to understanding Harry’s feelings and knowing instinctively what Harry needed to hear. And since James and Harry have different personalities and very different problems, I don’t think Sirius would be as helpful if he was just thinking of Harry’s personality as a carbon copy of James. His ability to connect with Harry shows an understanding of Harry as an individual.
If you’re married with a child and your spouse dies, I think it’s possible to find comfort in the personality/physical similarities between your kid and your deceased spouse without necessarily viewing the child as a substitute.
It’s hard to guess Sirius’s ultimate feelings. I’m sure he saw shadows of his best friend in Harry’s mannerisms and gestures, and I can see that being very comforting to him. However, I’m not sure he ever saw teenage Harry as a replacement for his old best friend. If Harry/Sirius had had another decade together, I can see the relationship transitioning into a more peer/peer relationship. But that’s a long way off. Canon Sirius did not view Harry as a peer, he viewed him as an underage teenager for whom Sirius was responsible.
To say that Sirius loved Harry because he was some sort of paltry facsimile of James is to suggest that Sirius only cared for Harry insofar as he replaced James in Sirius’s life. And that’s simply not true.