Well, it could be due to his appearance. Maybe he was wearing something similar to Hawkmoth's style by accident. Or perhaps his actions were misinterpreted. He might have been in the wrong place at the wrong time, like near a recent Hawkmoth attack site looking suspicious.
If Adrien were Hawkmoth in fanfic, it could change the dynamics of the story completely. It might explore his internal conflicts and the impact on his relationships with other characters.
Perhaps Adrien becomes Hawkmoth because he's been pushed to the edge by Ladybug's rejections. As Chat Noir, he's constantly in love with her but feels unappreciated. The pain of this unrequited love could twist his perception of good and evil. He might think that if he were in control as Hawkmoth, he could make Ladybug see him in a different light. He could use the power to create situations where Ladybug has to rely on him more, or at least understand his pain.
One implication could be that it might strain their friendship a bit. If Adrien really believes this in the context of fanfiction, he might start to act differently around Nino.
If Adrien becomes Hawkmoth in fanfiction, Paris would be in for a major shock. His friends, like Nino and Alya, would never suspect him at first. And Marinette, who is Ladybug, would be facing a completely different kind of enemy. She's used to fighting Hawkmoth as this unknown figure, but now it's someone she knows so well. It would change the dynamic of their battles entirely.