He becomes more nurturing. Dumbledore, who was mainly seen as a wise leader, now has to be a caregiver. He might show more patience and tenderness as he takes on the role of a father to Harry.
One interesting plotline could be how Dumbledore helps Harry discover his true magical potential beyond what he learned at Hogwarts. For example, he might introduce Harry to ancient and rare magic that was long forgotten. Another could be Harry's relationship with the wizarding community changing as he is now seen as Dumbledore's adopted son. This could lead to new friendships and alliances. Also, Dumbledore might train Harry to become a great leader in the wizarding world, preparing him for future threats that are much larger than Voldemort.
Harry becomes more confident. With Dumbledore as his adoptive father, he has a stronger support system. He is no longer as alone as he was, and this newfound security allows him to take more risks in his magical studies and in fighting dark forces. His sense of identity also shifts. He starts to see himself as part of Dumbledore's legacy, which gives him a greater sense of purpose.
In 'Dumbledore adopts baby Harry fanfiction', an interesting aspect could be how Dumbledore chooses to educate Harry. He might introduce him to different magical cultures and theories much earlier than in the original story. Also, the relationship between Dumbledore and the other Hogwarts staff when they find out he has adopted Harry. Some might be supportive, while others might worry about the potential implications for Hogwarts and the wizarding world as a whole.
Dumbledore, as Harry's guardian in the fanfiction, would probably use his influence in the wizarding world. He could make sure that Harry is taught by the best teachers at Hogwarts. He'd also be there to offer advice and guidance, especially when Harry faces challenges like dealing with Voldemort or other dark forces. For example, he might teach Harry some advanced defensive magic privately to better protect himself.
Well, in some fanfictions, Harry might attack Dumbledore if he discovers that Dumbledore has been using him as a pawn in a bigger game all along. This feeling of betrayal could drive him to lash out. Also, if there is a prophecy that he misinterprets to mean Dumbledore is a threat to his future or the future of his friends, that could be a motive.
Snape can be a prominent character. Given his complex relationship with Dumbledore and Harry, his perspective could add a lot to the story. He might have some insights into Dumbledore's actions that Harry doesn't know about, and his reactions to Harry's blaming could vary from being scornful to having a moment of understanding. Luna Lovegood could also be a part of it. Her unique view of the world might make her offer some different viewpoints on the situation between Harry and Dumbledore.
Dumbledore might give Harry advanced spellcasting training. He could teach Harry some of the more complex and ancient spells that are not commonly taught at Hogwarts.
Hermione could be a main character. She might use her vast knowledge to help Harry and Dumbledore in their adventures. For example, she could research ancient spells or potions that are crucial for their mission.
Perhaps there are safety concerns for the students that Dumbledore is not addressing adequately. Harry, having a strong sense of responsibility for his fellow students, may feel that the only way to ensure their safety is to remove Dumbledore from his position. He might believe that Dumbledore's actions or inactions are putting the students at risk, and thus he takes matters into his own hands.