Well, perhaps it's a planet with its own system of magic. It might have a core made of pure magic essence which gives power to all the magic on the planet. And there could be different regions, like one area filled with floating islands where powerful wizards live.
Perhaps he discovers the planet during one of his adventures. He could be using a new spell that accidentally transports him to this uncharted planet, and then through some series of events, he becomes the rightful owner. For example, he might save the planet from a great evil and as a reward, the planet's native beings make him the owner.
Perhaps it's an island with a hidden wizarding school, similar to Hogwarts but with its own distinct features. There might be ancient spells protecting the island, and the architecture could be a blend of different magical styles. Harry could have discovered this island during one of his adventures and decided to make it his own. He might use it as a sanctuary for those in need or as a place to study and preserve rare magic.
It could be a planet where magic is more powerful and less restricted. Harry, having faced so many limitations in the wizarding world he knew, would create a place where magic can be used freely for good. There could be no dark wizards or threats like Voldemort, and wizards could focus on exploring and expanding their magical abilities. For example, they could create new spells without the fear of them being used for evil.
In the fanfiction, he might own a building that is actually a magical library. Since Harry has always been interested in knowledge, this building could be full of rare and ancient magic books. It could also be a place where wizards and witches from all over come to study and research, and Harry, as the owner, would be in charge of protecting these precious knowledge resources.
Harry may decide to have a more intuitive magic system. Instead of relying on wands, which can be lost or broken, the magical beings on his planet could use their minds and emotions to cast spells. The stronger the emotion or intention, the more powerful the spell. And to prevent misuse, there could be a natural balance in the planet's magic. If someone tries to use dark magic, the planet itself could resist and weaken the spell, teaching the user that dark magic is not welcome.
There's also a possibility that it's a Chinese Fireball. These dragons are very beautiful and powerful. In a fanfiction, the author might choose this type of dragon for Harry Potter to own because it would add an element of exoticism and grandeur to the story.
Well, it could be a magic item shop. Since Harry Potter is from the wizarding world, he might sell things like wands, potions, or magical artifacts in Konoha.
It could mean he has the rights to certain fan - created stories related to Harry Potter. Maybe he's a collector of unique fanfiction works, or perhaps he's in a position to manage or curate a collection of such writings.
Well, perhaps Harry uses his magic to shape the planet. He might create unique magical creatures and ecosystems. For example, he could make a forest where the trees are made of living fire that doesn't burn but gives warmth and light.
Hermione would be a great help. Her vast knowledge of magic and books could provide the necessary information for building the planet, like the right spells and magical theories.