Percy's anger could make his powers go a bit out of control. For example, if he's near water, the water might start churning violently as he gets mad while protecting Annabeth. He could also become more physical in his protection, standing in a very aggressive stance in front of her, almost like a wild animal guarding its mate.
When Percy is protecting Annabeth in fanfic and is angry, he might become more impulsive. Instead of thinking things through, he just acts on his rage. For instance, if someone insults Annabeth, he might immediately lunge at them without considering the possible consequences, because his anger at the insult to her is so intense that all he can think about is defending her honor.
He might start yelling. Percy is known for his passionate nature, so when angry on the Argo 2, he could raise his voice at those who he thinks are at fault.
Well, it could be shown through her getting jealous easily when other girls interact with Percy. For example, if a new girl in the story tries to talk to Percy, Annabeth might give the girl a cold stare or quickly insert herself into the conversation.
In the fanfiction, Percy's possessiveness might also be demonstrated by his need to be included in all of Annabeth's plans. He may feel left out or threatened if she starts making plans with other people without him. He wants to be a part of everything she does, almost like he owns a part of her time and attention.
In a fanfic, there could be a new powerful enemy who targets Annabeth specifically. Percy gets mad that someone would dare to go after her. He immediately jumps into action, using his water powers to create a barrier around her and then engaging the enemy in a fierce battle, showing no mercy as he is so angry about the threat to Annabeth.
Love and loyalty are very common themes. Percy's actions are often driven by his deep love for Annabeth, and his loyalty to her as a friend and more. It's about him being willing to put himself in harm's way for her.
One popular 'Percy protects Annabeth' fanfic could be 'Percy's Vow'. In this fanfic, Percy makes a solemn promise to always keep Annabeth safe. It shows Percy using his powers and quick thinking to fend off various threats that come their way. Another might be 'Guardian Percy' where he takes on the role of Annabeth's personal protector during a dangerous quest. The author beautifully depicts Percy's determination and loyalty in safeguarding Annabeth.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
Often, she becomes overly possessive. If she sees Percy talking to another girl, she'll quickly insert herself into the conversation. She might put her arm around Percy or give the other girl a look that clearly says 'he's mine'. This possessiveness is a clear sign of her jealousy in the Percy Jackson fanfiction world.
Another one is 'Protecting My Athena' on Archive of Our Own. This one focuses a lot on Percy's internal thoughts while he is protecting Annabeth. It shows how his instincts kick in and how he plans his strategies to keep her out of danger, while also dealing with the normal challenges of their demigod lives.
Percy might develop more of a steely determination. In Tartarus, he has to face extreme danger, which could make him even more resolute in protecting Annabeth and getting out. Annabeth, on the other hand, could become more resourceful. She'd have to use her wits in this strange and deadly place.