Perhaps he's afraid of being used. If others know he's highly intelligent, they might try to take advantage of him in the fanfiction plot. He could also be hiding it as a form of self - protection. Maybe there are enemies who would see his intelligence as a threat and he doesn't want to put himself in danger.
One reason could be that he wants to avoid unwanted attention. In the fanfiction world, being overly intelligent might make him a target. Another reason might be that he is trying to fit in with a certain group that values other traits more. For example, if he's in a group that values physical strength over intelligence, he may hide his smarts to seem more like them.
He might want to protect himself. If others know how smart he is, he could become a target. Enemies might see him as a greater threat and go after him more aggressively.
He might pretend to not understand complex concepts. For example, when others are discussing advanced magic theories, he could act confused even though he actually gets it.
He may also choose to stay quiet during discussions where his knowledge could shine through. Instead of speaking up and showing off his intelligence, he lets others do the talking and only chimes in when it's necessary or when he can play dumb and ask simple questions.
Maybe he is afraid of being bullied. Smart kids sometimes get picked on by others who are jealous.
One reason could be fear. Maybe Harry discovers something about Dumbledore in the fanfiction that makes him scared, like a hidden agenda. Another reason might be to protect Dumbledore. Harry could think that by hiding from him, he is keeping Dumbledore away from some sort of danger that is related to Harry himself. Also, it could be due to a misunderstanding. Harry might misinterpret something Dumbledore does or says, and as a result, decides to hide from him.
One reason could be that he doesn't want his friends to worry. Harry often takes on a lot of responsibility and might see his sickness as a burden he doesn't want to place on others. Another reason could be related to his enemies. He might fear that if Voldemort or his followers find out he's sick, they'll see it as a weakness and target him more aggressively.
Maybe he wants to start anew. Fanfic often has its own set of pre - conceived notions and storylines associated with a particular name. Changing his name legally allows Harry to break free from those and create his own identity outside of the fan - created narratives. It could also be for protection. There might be some negative aspects in the fanfic world related to his name, such as being associated with a certain group or plotline that he doesn't like, so changing his name is a way to protect his reputation.
She could be hiding from those who want to harm her. In the wizarding world, there are always threats, and hiding as an animagus is a clever way to stay safe.
Maybe he wants to blend in better. If he's in a situation where he needs to gather information undercover, showing off his skills would blow his cover. So he hides them to seem like an ordinary person until the right moment to strike.
One reason could be that he doesn't want to stand out too much. In the fanfiction world, just like in the original Harry Potter story, being overly intelligent might draw unwanted attention, especially from the bad guys. He might be afraid of being targeted more if his intelligence is known.