In this fanfiction, perhaps the dog that bit Snape was under some dark magic influence. Harry not only cleans the physical bite but also has to figure out the source of the dark magic. He might have to search for clues in Hogwarts, and along the way, he and Snape start to understand each other better.
Perhaps it's one of the Hogwarts students. The dog might have been provoked by the student's actions, and Snape is too embarrassed or proud to clean the bite himself, so Harry is made to do it.
It could be an experienced fanfic author who has written a lot about the Harry Potter series. They might have decided to write this particular story to add a different kind of interaction between the two characters. Maybe they noticed that there are not many stories about Harry helping Snape in such a specific way.
Well, perhaps Harry discovers Snape's dog has bitten someone. He might initially be reluctant to get involved but his sense of justice makes him step up. He could start by finding out how severe the bite is and then look for a way to clean the wound properly to prevent infection.
The plot could be that Dumbledore assigns Harry to clean Snape's home as a way to build a bond between them. Harry initially resents it but as he cleans, he finds a diary that reveals Snape's unrequited love for Lily. This discovery makes Harry see Snape in a new light and he decides to try and befriend him. He starts to clean with more care, and when Snape returns, he is initially angry but then realizes Harry has a newfound respect for him.
The plot might start with a magical experiment gone wrong. Harry ends up with a special mark or power that makes him Snape's master. As the story progresses, Harry uses this power to make Snape teach him advanced potions and dark arts defenses. Snape resents it but has no choice. Meanwhile, Voldemort senses this strange new power dynamic and tries to use it to his advantage. This forces Harry and Snape to work together in ways they never thought possible to outwit Voldemort while also dealing with their own difficult relationship.
Maybe in the fanfiction, Voldemort devises a plan where Snape forgets Harry to disrupt the Order of the Phoenix's plans. Without Snape's knowledge of Harry and his role in the prophecy, things start to go awry. Harry has to find a way to make Snape remember, perhaps by going on a quest to find a magical relic that can restore memories. Hermione and Ron would be by his side, of course, providing their usual support and knowledge.
In this fanfiction, perhaps Harry is in a difficult situation where Snape has some form of control over him. It could be related to a school task gone wrong, like a potions experiment that Snape blames Harry for. And Harry is struggling to break free from Snape's disciplinary grip.
Well, perhaps Harry could be transformed into an animal form that is Snape's familiar. Maybe he ends up having to follow Snape's commands in this new form and through that, they start to understand each other better. For example, Snape might initially be cold to Harry even in this situation but as time passes, he realizes Harry's loyalty in his new role as a familiar.
One possible plot could be that Snape, as Harry's uncle, tries to secretly train Harry in advanced magic. He does this because he wants Harry to be strong enough to face Voldemort, but he has to keep it a secret from Dumbledore and others in Hogwarts who might not trust his motives.
Well, one possible plot could be that Snape uses some kind of powerful magic to transform himself into a female form and becomes Harry's mother. He could have done this to protect Harry in a more direct way, perhaps knowing some dark prophecy about Harry. And then the story could follow Harry's growth with Snape as his mother, dealing with the secrecy of it all.