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.
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.
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.
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.
Perhaps he uses the island as a secret training ground. He could practice advanced spells away from the prying eyes of Hogwarts faculty and students. Harry might also build his own magical fortress on the island for protection and privacy.
There could be some of the house - elves that Harry has freed. They might choose to live on the island, using their magic to help with various tasks such as farming magical plants or taking care of the island's infrastructure. Also, some of the students from Hogwarts who were inspired by Harry could be there. They might be learning new forms of magic that are specific to the island or studying the unique magical ecosystem there.
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.
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, in some fanfictions, it could be due to a powerful spell gone wrong that transfers the ownership to him. Maybe a spell meant to protect Diagon Alley's property rights backfires and suddenly Harry is the owner.
In this fanfic, perhaps Harry could turn the island into a magical sanctuary for endangered magical creatures. He might build a school there for young witches and wizards who are overlooked by the mainstream magical education system.
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.