Perhaps the obsessed guy has a change of heart. He realizes his actions are wrong and lets the girl go. But this ending often comes with a complex set of emotions and consequences for both characters as they try to make sense of what has happened.
Often, the theme of rescue is present. It could be the girl trying to rescue herself or her friends and family attempting to rescue her. Also, there can be a theme of psychological manipulation as the obsessed guy may use various tactics to control the girl.
They're popular because it's a very relatable concept. Many people have had a crush on someone and felt that sense of obsession at some point. It's also a bit of a fantasy for some, seeing someone be so dedicated to another person.
Well, often in these stories, the guy starts as a bit of a stalker - not in a creepy way though. He'll find out her daily routine and 'accidentally' bump into her a lot. Then there's the plot where he saves the girl from some sort of trouble, which makes her see him in a new light. Also, there's the scenario where the guy writes her love letters or poems to express his obsession in a more romantic way.
Once upon a time, there was a young girl named Lily. One day, as she was walking home from school, a strange van pulled up beside her. Before she could react, a masked man grabbed her and dragged her into the van. Lily was terrified. She was taken to an old, abandoned warehouse. There, she tried to find a way to escape. She looked around for any possible tools or exits. While the kidnappers were busy talking among themselves, Lily found a small, rusty nail. She used it to try and pick the lock on the door she was locked behind. After several attempts, she finally managed to open the door and ran as fast as she could. She found a road and followed it until she reached a small town where she asked for help and was eventually reunited with her family.
One possible plot is that the guy in the van is actually under the control of a more powerful villain. The anime girl, who might be a key figure in the anime's lore, is kidnapped to be used as a bargaining chip. However, during her captivity, she befriends a small creature or another captive in the van. Together, they come up with a plan to escape and expose the real villain behind the kidnapping.
She may start by trying to figure out who the kidnapper is and what his motives are. If she has any special abilities or skills, she would be thinking about how to use them to her advantage. For example, if she has telepathic powers, she could try to send a distress signal to her friends or family. Also, she might try to gain the kidnapper's trust in order to find a moment to escape.
One common problem is the ethical aspect. Kidnapping is a serious crime and presenting it in a way that might be romanticized in fanfiction is not okay. It can give wrong ideas about relationships and boundaries.
One sign could be that he becomes very secretive about his whereabouts. In a fanfic, if he used to be open but suddenly starts making excuses not to say where he's been, it might be a red flag.
Rather than seeking out this specific and potentially concerning type of fanfiction, you could explore fanfiction in genres that are more widely loved and appropriate. For example, there are great fanfictions about characters growing and learning in a fictional world, or ones that explore the backstories of beloved characters in a positive light.