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.
There might be some sort of magical bargain or prophecy involved. For example, a long - forgotten prophecy stated that Harry Potter would one day own the bank to bring about a great change in the wizarding financial system, and through a series of events, this prophecy is fulfilled.
He might introduce more fair lending practices. In the wizarding world, some families or individuals might have been unfairly treated when it comes to borrowing money from the bank. Harry could change that.
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.
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.
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.
The characters would first need to identify the robbers. Harry Potter has his Invisibility Cloak, so he could use it to spy on the robbers and find out their plans. Once they know what's going on, they'd gather their allies. Dumbledore, if he were still alive in the fanfiction, would offer his wisdom and powerful magic. The trio - Harry, Hermione, and Ron - would then work together with other Hogwarts students or members of the Order of the Phoenix. They might set up magical traps around the bank to catch the robbers or use spells to immobilize them when they make their move.
They could be used to create a mystery. Say a set of bank statements has some encrypted or missing information. Harry and Hermione would use their knowledge of magic and codes to try and decipher it, leading them to a secret vault or a hidden treasure.
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.