Let's say you become a deer. Your character could develop the ability to sense danger from afar. This could be because deer in the Naruto world have a natural instinct for self - preservation. You could also develop the ability to blend into the forest better, like using your antlers to create a kind of camouflage. Another way to develop abilities is by making pacts with other creatures or spirits. If you make a pact with a water spirit, you might gain the ability to control water in some way, even if you're just a simple forest deer.
If you become an animal in this fanfiction, your character could develop new abilities through training. For example, if you're a wolf, you could train to run faster and have better endurance in the wild. You might also learn how to communicate with other animals in the forest, which could be a useful ability for gathering information.