They usually introduce new animal species or expand on the existing ones in the Naruto world. Like there could be some rare, powerful animals that only Naruto can befriend. These animals might have unique abilities such as the power to heal or to create illusions on a large scale.
Power exploration is also a big theme. Writers often take the opportunity to create unique and powerful abilities based on the animal bloodline. They might describe how a character can transform parts of their body into an animal form, like having fox ears and a tail for a character with a kitsune bloodline, and how they learn to control and use these abilities in combat or in daily life. And this exploration of power can lead to some really exciting action scenes in the fanfiction.
Often, they emphasize the loyalty between Naruto and his ninken. For example, in many such fanfictions, the ninken will stand by Naruto no matter how difficult the situation is.
Typically, in 'Naruto has augments' fanfiction, Naruto gets some sort of extra abilities or enhancements. This could be physical like super strength or speed, or it could be related to his jutsu, making them more powerful. For example, he might gain the ability to control an element more efficiently or in a new way.
Typically, in 'naruto has foxfire' fanfiction, Naruto's power gets an extra dimension. The foxfire is often related to his connection with the Nine - Tails. It might be a new form of his chakra, or a special ability he discovers.
In some fanfictions, the nin animal could have a connection to an ancient prophecy. So Naruto has to figure out how to use his nin animal in relation to this prophecy to save the village. Maybe the nin animal has a hidden form that is only revealed when certain conditions are met.
One popular Naruto fanfiction with animals could be 'Naruto and the Summoning Menagerie'. It often features Naruto forming unique bonds with various summoned animals in ways not shown in the original series.
Obviously Naruto is one of the main characters. His nin animal would also be a main focus. There might also be his friends like Sasuke and Sakura who interact with Naruto and his nin animal in different ways. Maybe Sasuke is initially jealous of Naruto's unique nin animal.
Typically, in such fanfiction, Naruto is likely to be at the center of the story. The 'puppet' aspect might involve characters related to the puppet - using techniques in Naruto, like Kankuro perhaps. The 'harem' part means there will be multiple characters, usually female, having some sort of relationship with Naruto. It could be romantic, or they might be allies in a unique way.
A dog would suit Naruto's personality well. Just like Naruto is loyal and friendly, dogs are known for their loyalty. They could be by Naruto's side through thick and thin, and Naruto would love having a furry friend that's always excited to see him. Also, dogs can be trained in various skills that could be useful in the ninja world, like tracking enemies or guarding a location.
In Naruto fanfiction, a lot of creative animal partners can be found. For example, there could be a magical dolphin that can swim through the air as well as water. It could be used to quickly transport the ninja across long distances. Another idea is a phoenix - like bird that can resurrect from its ashes and provide protection to its partner in the form of a fire shield. And then there's the possibility of a bear with super strength and the intelligence to strategize in battles.