How did Harry Potter come to own the castle in the fanfiction?

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.

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 he received it as a gift from another fanfiction enthusiast who knew his interest in Twi'leks.

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.

There's a possibility that after Sirius's death, Harry just sort of took over. He had already spent time there and knew the place well. The remaining members of the Order of the Phoenix, seeing his connection to the place and his leadership potential, supported his claim to Grimmauld Place.

He could use the money left to him by his parents. They were wealthy and it's likely there was a large inheritance waiting for him. Plus, he had his own earnings from winning the Triwizard Tournament.

She might have found it online. There are numerous websites dedicated to Harry Potter fanfiction.

The castle might be a grand and ancient structure. It could have high turrets reaching towards the sky, with thick stone walls that have withstood the test of time. There might be a large moat around it, filled with mysterious creatures. The interior could be filled with secret passages and rooms, just like Hogwarts. There could be a great hall with a long table for feasts and a fireplace that always has a warm fire burning.