One common theme is protection. Sirius feels an overwhelming need to protect baby Harry, given what has happened to his parents. Another theme is the idea of family. Even though Sirius isn't Harry's father, the bond between them is portrayed as very strong, like a family bond. And there's often a theme of hope as well, that with Sirius in Harry's life, there's a glimmer of a better future for Harry.
It's popular because it offers a different take on the relationship. Sirius is a complex and beloved character, and seeing him interact with a baby Harry is both cute and full of potential for interesting stories.
Well, at first, Sirius would be in a state of shock and panic. But then he'd have to step up. Their relationship would shift from one of equals (in a sense, as Harry was growing up) to a caretaker - cared - for dynamic. Sirius would have to learn about Harry's likes and dislikes as a baby. He'd sing lullabies to him, comfort him when he cries. This new experience would make their bond stronger in a very different way. Harry might not remember all of it when he's turned back, but there would be a subconscious change in how he feels towards Sirius, a deeper sense of security and love that stems from those baby - days when Sirius was his whole world.
Sirius could use his knowledge of the wizarding world's secrets. He might know some hidden places or spells that could keep Voldemort at bay. For example, he could put a powerful protective charm on the place where Harry is staying.
A third one is 'Sirius's Adventure with Baby Harry'. In this story, Sirius takes baby Harry on little adventures around the wizarding world. They visit different magical places and meet interesting characters. It's a story that emphasizes their relationship and how it grows over time.
One common theme is sacrifice. Sirius is often willing to put his own life on the line to save Harry. For example, in many fics, he battles powerful dark forces without hesitation to protect the baby.
Sirius would probably be very protective. He'd use his magic to create a safe and comfortable environment for Harry. For example, he might conjure up soft, warm bedding for the baby. He'd also try to find the best baby - wizarding supplies, like magic - infused milk bottles that ensure Harry gets all the nutrition he needs.
Sirius influences Harry's early life by being a protector. He stands as a shield against any potential threats. He might use his Patronus, a large dog, to guard Harry at night, making Harry feel safe in his sleep.
Well, Sirius might see some of his own qualities in Harry's boyfriend. Maybe the boyfriend is brave like Sirius was, and Sirius would then start to warm up to him. He could share some of his old Hogwarts stories with the boyfriend, which would help the boyfriend understand Harry's world better.
In many fanfictions where Sirius meets Remus' parents, it can be a very interesting scenario. It might be filled with tension as Sirius is often seen as a bit of a rebel. Maybe at first, Remus' parents are cautious of Sirius, given his family's reputation. But as they get to know him, they could see his true self and how much he cares for Remus.