One common theme might be the struggle against societal norms. Since this kind of relationship is not acceptable in the normal world, the fanfiction could explore how the characters deal with the outside world's judgment. But this is still a very controversial theme.
Romance between Harry Potter and his sister goes against the original canon and can be considered quite abnormal. However, if you still want to explore fanfiction in this regard, you could try searching on some large - scale fanfiction websites like Archive of Our Own. But be aware that such content may not be in line with the general moral and ethical concepts of the original work.
One possible reason could be jealousy. Maybe the sister is getting more attention from their parents or at Hogwarts. For example, if she is a star student and Harry feels overshadowed.
Obviously, Harry Potter himself is a central character. His sister would be the new addition. Hermione and Ron are likely to be there as well. They are Harry's best friends in the original series and would probably play important roles in the fanfiction too. Dumbledore might also appear, as he is a very important figure in the wizarding world and could guide the sister in her magical journey.
If such fanfiction exists, it may lead to a lot of negative impacts. It can mislead some young and inexperienced readers into thinking that abnormal relationships are acceptable. Moreover, it might face legal issues in some cases as it violates ethical and moral boundaries. Also, it will be shunned by the vast majority of the legitimate Harry Potter fandom who uphold the positive values of the series.
Hermione would still be a very popular character. She could be the one to help the twin sister study and understand the more complex aspects of magic. Her logical thinking and vast knowledge of magic would be invaluable in their adventures.
Their parents' old friends could be main characters. They might come to check on Harry and his sister, bringing gifts and stories about their parents. These friends could offer guidance and protection when needed.
Ron and Hermione are likely to interact with them. Ron might be a bit more protective of the sister, while Hermione would probably study and share knowledge with her. They could all be in the Gryffindor common room, discussing spells and strategies.
A more complex plotline could involve the sister being targeted by Voldemort because he sees her as a threat to his plans. She might have a prophecy associated with her that predicts she will be the one to finally defeat him. This would put her in a lot of danger and force Harry to be even more protective of her while also relying on her unique skills to fight against the Dark Lord.
Voldemort's remaining followers could be antagonists. They might target Harry's little sister thinking they can use her to get to Harry. They could try to kidnap her or use dark magic on her.