Her relationships would be different. Instead of having school friends like Harry and Ron, she might form alliances with other self - taught wizards or those from different magical backgrounds. This could lead to a more diverse and complex social network, which in turn would shape her character in unique ways. For instance, she could become more diplomatic as she deals with different magical cultures outside of the Hogwarts - centric wizarding world.
Perhaps she discovers a different magical school or a secret magical community outside of the Hogwarts system. In this fanfiction, she could be mentored by an eccentric and powerful wizard who has his own way of teaching magic, far removed from the traditional Hogwarts curriculum.
Her character could develop through her relationships. She might befriend students from all houses, which is a bit unusual in Hogwarts. This would be because she has qualities from both Gryffindor (from Hermione) and Slytherin (from Snape). Through these friendships, she'd learn different perspectives and grow as a person. For instance, she could teach her Gryffindor friends about the subtleties of Slytherin thinking and vice versa.
Her character might develop in a way that she constantly tries to prove herself. Because of her parents' polar - opposite reputations, she would feel the need to show that she is not defined by their actions or reputations. She could excel in both academics and magic - related sports, like Quidditch, to gain respect.
One possibility is that Harry grows up in the Muggle world completely unaware of his wizarding heritage. He might lead a normal life, going to a regular school, making Muggle friends, and having no idea about the magical world that could have been his. Maybe he has a natural aptitude for something like martial arts or a particular academic subject that in the wizarding world would be replaced by magic.
She might become more independent. Without the structure of Hogwarts, she'd have to figure things out on her own. For example, she'd have to find her own sources of magical knowledge. This could make her a bit of a maverick in the wizarding world.
He'd likely be less jaded. At Hogwarts, he faced a lot of hardships like being bullied by the Marauders. Without that, he might be more open and less defensive. His social skills could be better as he wouldn't have developed the same level of distrust towards others.
One possible plot could be that Harry Potter discovers a different magical school. Maybe he stumbles upon a hidden school in the Muggle world that has been secretly teaching magic. He could form new friendships and face different types of magical challenges there. It would be a completely new magical experience for him, away from the familiar Hogwarts environment.
The Dumbledore character could still be important. He might be secretly watching over Harry, guiding him from a distance. Another character could be a new wizard or witch that Harry meets in the Muggle world who has a special mission related to Harry. For example, a young witch who has been exiled from the wizarding world but has the knowledge to help Harry understand his powers. And of course, Voldemort would still be a main antagonist as his threat looms over the entire wizarding world whether Harry is at Hogwarts or not.
In post - Hogwarts fanfiction, Harry often develops into a more confident leader. He has already defeated Voldemort, so he uses his experience to mentor the next generation. Hermione, on the other hand, may become more assertive in the academic and political aspects of the wizarding world.
Well, without Ron and Hermione, Harry's character would lack the moral compass that Hermione often provides. He might be more impulsive and make decisions without fully thinking them through. For example, in the original story, Hermione stops Harry from using the Marauder's Map too recklessly. Without her, Harry could get into a lot more trouble with it.