No, Twilight was not a fanfic of Harry Potter. They are two completely separate and distinct literary works with their own unique storylines and characters.
No. Twilight and Harry Potter are two completely different series. Twilight focuses on vampires, werewolves, and a human girl's relationships in a supernatural world, while Harry Potter is centered around magic, wizards, and a boy's fight against the dark wizard. There's no evidence that Twilight began as Harry Potter fanfic.
One common theme is the clash of different supernatural elements. Since Harry Potter has magic, wizards, etc., and Twilight has vampires and werewolves, seeing how these interact is a big theme.
In fanfics, the red string of fate can be a creative device. For Harry Potter, it could be a magical string that only certain magical beings can see. It might show a connection between a wizard and a magical creature. In Twilight, it could be something that the Volturi are aware of and use to manipulate relationships. For instance, they could try to sever the red string between a vampire and a human to stop a potential powerful union.
When it comes to plot elements, a similarity is that both Harry Potter and Twilight have a sense of mystery. In Harry Potter, there are mysteries about Hogwarts, the Deathly Hallows, etc. In Twilight, there are mysteries about the vampire abilities and the history of the clans. But the pacing is different. Harry Potter has a faster - paced plot with a lot of action - packed events, like the Triwizard Tournament and battles against Death Eaters. Twilight has a slower - paced plot that focuses more on the emotional and relationship - building aspects, such as Bella's slow - burning relationship with Edward and her friendship with Jacob.