Yes, he is. Fanfiction often delves deeper into the idea of Severus Snape's connection to Hogwarts. Since he dedicated his life to the school in different ways in the original story, fanfic creators expand on this. They might write about him being magically bound to Hogwarts, where he can't stray too far from the castle grounds. This could be because of a spell he cast himself, or as a result of some ancient magic related to his role at the school.
One possible reason in fanfiction could be that Snape's constant bullying and unfair treatment finally pushed Harry over the edge. Harry might have felt that he couldn't take it anymore, especially if Snape made his life at Hogwarts unbearable in ways that the other teachers didn't notice or couldn't stop.
People write it because Snape is such a complex character. His leaving in the original story leaves a lot of room for speculation. Writers want to fill in the gaps about what he might do next, where he might go, and what his future holds.
There's also 'Snape's Silent Exit Continued'. This fanfic might focus on how his leaving affects the other characters. For example, how Harry Potter really feels about Snape's departure when he finally has time to think about it, and how it changes his view of Snape in the long run.
In fanfiction, one reason could be his desire for a fresh start away from the memories at Hogwarts. Maybe he feels he has done all he can there and wants to explore new magical knowledge elsewhere.
The Hogwarts staff would also be affected. There could be a shift in power dynamics. Maybe some characters who were in his shadow get a chance to shine, while others might struggle to fill the role he played in maintaining a certain level of order and mystery in the school. For example, his departure could make it easier for some to implement new teaching methods, but at the same time, it could also expose weaknesses in the school's security as he was very knowledgeable about the Dark Arts.
Snape would likely become more confident in his place at Hogwarts. As the heir, he has a new - found sense of belonging that could soften his harsh exterior. He might be more open to helping students, especially those who show an interest in the history of Hogwarts.
One way could be through a magical oath. In some fanfictions, Snape might have sworn an oath to protect Hogwarts at all costs, which binds him to the school. Another way is through his love for Lily Potter. Since Hogwarts was a place that was important to her too, his emotional connection to her could translate into a binding to the school. Also, he could be bound by the magic of the place itself. Hogwarts is full of ancient magic, and it might recognize Snape's loyalty and determination, thus binding him to it.
One possibility could be that Snape grows up in a different magical environment, perhaps in a small, hidden magical community that has its own unique way of teaching magic. He might develop his skills in potion - making and the Dark Arts in a more self - taught or community - guided way. Without the influence of Hogwarts, his relationships would be very different. He may not have the same intense rivalries or unrequited love for Lily Potter.
An interesting plot might be Snape's journey to fully understand what it means to be the heir of Hogwarts. He could start having strange dreams or visions related to the founders of Hogwarts. As he delves deeper, he uncovers secrets about the school's past that not only change his view of Hogwarts but also of himself. This new knowledge could also impact his relationships with the students and other teachers, as he realizes he has a greater responsibility towards the school than he initially thought.