A RWBY character with the Sword of Gryffindor might also experience a change in their personality or outlook. Since the sword is associated with Gryffindor values, the character could become more courageous and noble. They might also start to uphold values that are more in line with the Harry Potter universe, which could create an interesting dynamic within the RWBY world.
Often, it could be a plot device. For example, it might be the key to unlocking an ancient mystery in the RWBY world. Since the Sword of Gryffindor only appears to those worthy in the Harry Potter series, in RWBY fanfic, it could be that only a truly noble character can access it. And when they do, it causes a stir in the RWBY world as it's an unknown and extremely powerful object.
Yes, in some fanfictions, Harry Potter is depicted as the heir of Gryffindor. It adds an interesting twist to the original story. The connection to the Gryffindor sword becomes more profound as it is often a symbol of Gryffindor's legacy. In these fanfictions, Harry might discover hidden powers or a deeper understanding of his destiny through this new identity.
The dragon sword in the fanfiction might be long and slender, with a blade that shimmers like dragon scales in the sunlight. The hilt could be made of a dark, enchanted metal, perhaps with a dragon's head carved at the end, its eyes seeming to glow with an inner fire.
It could be a symbol of Gryffindor's spirit in the fanfic. When Harry wields the flame, he is representing all the brave and noble qualities of the Gryffindor house, and it might inspire others around him to be braver too. Also, the flame might have some special abilities like being able to break dark spells or protect the innocent.
The dragon sword might give Harry Potter enhanced combat abilities. It could potentially allow him to cut through dark magic defenses more easily, much like how the Sword of Gryffindor could destroy Horcruxes. Maybe it gives him the power to communicate with dragons, since it's a dragon sword, and use their strength in battles against Voldemort and his Death Eaters.
Maybe he stumbles upon it during a time of great danger in the fanfic. For example, when facing a powerful dark wizard, he might feel an unusual warmth and strength within him, and then realizes it's the power of the Gryffindor flame.
It could have the ability to communicate with Harry on a psychic level. Telling him when danger is near or guiding him on how to use its powers most effectively in different situations.
There could be many reasons. Maybe some fanfiction writers want to create an alternate universe where Harry has a different path. It could also be for the sake of creating a more complex and unexpected plot, like having him be falsely accused of something so bad that Gryffindor doesn't want him. Another possibility is that the writer wants to explore how Harry would react and grow outside of the Gryffindor environment he's so closely associated with in the original series.
One possible reason could be a false accusation of using dark magic. In some fanfictions, the wizarding world can be very paranoid about dark magic, and if Harry was wrongly suspected of delving into it, Gryffindor might exile him to protect their reputation.
One possible plot could be that Harry is framed for something he didn't do. Maybe a powerful dark object is found in his dorm, and the Gryffindor house, fearing for their reputation and safety, decides to exile him. Harry then has to prove his innocence while also dealing with the isolation and new challenges outside of his familiar house.