Well, Naruto might show his prodigious side in a more extroverted way compared to Itachi. Itachi was a prodigy in his calm and calculated skills from a young age. Naruto could be shown as a prodigy in quickly mastering new jutsus or having an uncanny ability to lead and inspire others in the fanfic, much like Itachi was a natural leader in his own way among the Uchiha.
It could be a gesture of affection or a way to start a conversation. Sometimes in fanfictions, small actions like this have deeper meanings related to the plot or character development.
One reason could be for strategic purposes. Itachi was a highly strategic ninja. So, in fanfiction, Naruto might act like him to outwit a particularly difficult enemy. For instance, when facing an enemy who is a master of deception, Naruto could adopt Itachi's calm and calculating demeanor to see through the enemy's tricks.
Maybe in the fanfic, Itachi has done something that goes against Sakura's beliefs or values. For example, he might have made a decision that put her friends in danger.
Well, in 'Itachi raises Naruto' fanfictions, we get to see two very different worlds collide. Itachi has his own dark past and experiences with the Uchiha massacre, while Naruto is all about hope and becoming Hokage. When Itachi takes on the role of raising Naruto, it allows for a lot of character development for both. Itachi can pass on his knowledge and skills, and Naruto can show Itachi a different way of looking at things. Also, it gives the author a chance to explore different plotlines that wouldn't exist in the original story, like how Itachi would train Naruto differently from Jiraiya, for example.
Perhaps Itachi realizes that Naruto has the potential to change things for the better in the ninja world. He takes Naruto with him to train him in ways that the Leaf Village couldn't. Itachi may have some knowledge or skills that he wants to pass on to Naruto, like how to deal with the power of the Nine - Tails in a more efficient way.