Another reason might be related to his home situation. Although it's not as prominent in the fanfiction's context perhaps, in the original story, the Dursleys were not kind to Harry. If this aspect is incorporated into the fanfiction, it could be that he got the bruises from their mistreatment before coming to Hogwarts.
He might be initially shocked. Snape has this complex relationship with Harry. He may try to hide his concern behind his usual cold exterior, but his eyes might give away his surprise.
One possible reason could be his lingering resentment towards James Potter, Harry's father. Snape and James had a long - standing feud during their school days. Since Harry looks a lot like James, Snape might be transferring his old grudges onto Harry.
One possible reason could be Snape's long - standing grudge against James Potter, Harry's father. Since Snape had a difficult past with James, he might project that onto Harry. Another reason could be that Snape is trying to protect Harry in a very strange way. He may think that by rejecting Harry, he is keeping him at a distance from the dangerous wizarding world politics and plots that Snape is often involved in.
No, 'Bruises' is likely fictional. Many stories are created from the imagination of the author rather than being based on real events.
He could feel a sense of guilt. Snape has always had a role in Harry's life, sometimes as an adversary. Knowing that Harry has these scars which are a result of Voldemort's actions, and Snape being involved in some way with Voldemort in the past, he might feel responsible on some level.
One reason could be a surprise attack by Voldemort or his Death Eaters. Maybe they found a way to bypass all the protection spells around Harry and launched a sudden, fatal assault.
One possible reason could be that Snape shows a different side to Harry in the fanfiction. Maybe he protects Harry secretly from some hidden threats that others are unaware of, which makes Harry start to trust him.
Another theme might be mystery. We don't know how Lisa King got the 23 bruises from just the title. So the story could be about uncovering the mystery behind those bruises, which could involve elements of suspense and investigation. Also, it could be about resilience. Despite having 23 bruises, Lisa King may show great strength and resilience in the face of whatever caused them, whether it was an accident, an attack, or some other difficult situation.
Maybe Harry discovers something about his past, perhaps related to his parents, that only Snape can help with. Snape was closely associated with Harry's parents. He might know some secrets or have some information that could help Harry understand a family mystery or some hidden power within him.
One possible reason could be Snape's personal bias against Harry. Since Harry is James Potter's son, and Snape had a difficult relationship with James in their school days, Snape might unconsciously take out his old grudges on Harry.