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.
It could be that the dragon was an egg he found by accident. He, being curious and having a soft spot for magical creatures, took the egg and hatched it, thus becoming the owner of the dragon. In the wizarding world, there are often unclaimed or lost magical items and creatures, and Harry might have just stumbled upon this dragon egg.
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.
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.
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 uncovers some major corruption or inefficiency in the Ministry. So, as a result of his efforts to reform it, he is given control. For example, if he finds that the Ministry has been misusing magic for personal gains, he could lead a movement to take it over and run it in a more just way.
He might have discovered the island during one of his adventures and claimed it as his own. Since there was no one else on the island at the time, and he was the first to set foot on it and recognize its potential, he decided to make it his property.
Maybe he inherits it. In the wizarding world, there could be some old family connections or a will that passes the ownership of the bank to him.
In many fanfictions, Harry might use his powerful spells like the Stunning Spell or the Killing Curse if the story allows for such dark magic use. He could also use his intelligence to outwit the dragon, perhaps finding a weakness in its armor or behavior pattern and then using a well - timed spell to defeat it.
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.
Well, in some fanfictions, Harry Potter might come to own the Ministry of Magic through his heroic deeds. He could have defeated a great evil that threatened the wizarding world so severely that the wizarding community decides he is the most fit to take over and reform the Ministry.