One reason could be his pure heart. Harry's pure intentions and lack of desire for power for power's sake gives him an edge. His magic is not corrupted like Voldemort's. It flows more freely and is more in tune with the natural order of magic. So, when he casts spells, they are more effective.
It could also be due to some hidden magical heritage that is revealed in the fanfiction. Maybe he has a connection to an ancient and powerful magical line that Voldemort doesn't know about. This heritage could give him access to unique magics or make his existing spells much stronger, allowing him to surpass Voldemort in power.
Maybe in this fanfiction, Harry has a hidden power within him that is only revealed later. It could be something related to his family heritage that Dumbledore doesn't possess. Also, the author might have decided to give Harry more power to explore different storylines. For instance, Harry could be more powerful because he has a stronger bond with the magical creatures, which gives him additional powers that Dumbledore lacks.
In this fanfiction, it could be that Harry's power stems from his experiences. He has faced numerous dark wizards and challenges, which have forced him to grow stronger. Merlin, on the other hand, might not have had such intense and varied experiences. So, through his trials, Harry has developed powers that surpass Merlin's.
Harry may show greater power in his connection to the magical world around him. He has always had a special bond with Hogwarts. In the fanfiction, this bond could be enhanced. So, when Voldemort tries to attack the school, Harry is able to draw on the magic of Hogwarts itself. He can use the ancient magic stored within the castle walls to create powerful shields and offensive spells that are far more potent than anything Voldemort can muster.
One reason could be a prophecy misinterpretation. Voldemort might initially think that Harry is key to his success in a different way than he originally thought, and so he saves him to figure out this new angle. For example, if there was a prophecy about a 'chosen one' and Voldemort realizes he needs Harry alive to fulfill some part of it that benefits him.
One reason could be for comical effect. It's quite funny to think of the powerful and feared Voldemort as a pet. Another reason might be to explore a different power dynamic. If Voldemort is a pet, Harry has a certain level of control over him, which is a complete reversal of their normal relationship in the original story.
One reason could be for added protection. In fanfiction, the author might think that one Patronus isn't enough for all the threats Harry faces. Another reason could be to show his connection to different magical aspects. For example, if he has a Patronus related to a water creature and another to a forest creature, it could represent his ties to different elements of the magical world.
Well, in some fanfictions, Harry Potter might be seen as more powerful than Merlin because Harry has faced and defeated Voldemort. That's a huge accomplishment. Merlin, while very powerful in his own right, didn't have to deal with a dark wizard as complex and powerful as Voldemort. Harry's ability to survive the Killing Curse as a baby also gives him a sort of 'chosen one' status that could be seen as making him more powerful.
Magic knowledge. Harry has a lot of knowledge from Hogwarts, which could be more comprehensive than what Merlin had. So in fanfiction, this knowledge could be used to show he's more powerful.
In some fanfiction, Harry leaves to seek more powerful magic. He realizes that the magic in Britain is somewhat limited in the fight against Voldemort. So he travels to other places, like ancient magic - rich regions in Asia or Africa, to find a way to defeat Voldemort. He hopes that by leaving, he can gain the upper hand in the battle.
In Harry Potter fanfiction, some often considered powerful wizards could be the main characters with enhanced abilities. For example, a fanfic might make Harry Potter even more powerful by giving him mastery over new spells or magic that was not in the original books. Another could be Hermione Granger, perhaps with even greater knowledge of ancient magic, allowing her to perform feats that were beyond her in the official story. There are also original characters created in fanfiction that are sometimes made extremely powerful, with unique magical talents that set them apart from the regular characters in the series.