Not every anime warrior fights using only their own bodies. Whether it's to avoid damage to their own bodies or to utilize a set of skills that their human forms do not possess, there are some anime characters who fight with animals.
Anime characters with animal manipulation utilize their skills in a startling variety of ways. Some, like Shino Aburame of Naruto, have a symbiotic relationship with the animals that he battles with - in exchange for obedience, they get to feed off of his chakra. Others, like Kouji Kouda of My Hero Academia, can control animals to do their bidding and offer friendship in return.
These kinds of techniques can range from totally adorable to downright terrifying, depending on the animal in question and the exact nature of their abilities - Gomamon's March of Fishes technique is cute as heck, but Zouken Matou's worms will make your skin crawl, literally. Either way, they provide a unique spin on battle techniques.
There are actually a lot of Naruto characters who use animals in combat, but Shino is probably the most notorious. The entire Aburame family has a symbiotic relationship with kikaichu, beetle-like insects that inhabit their bodies and feed off of their chakra in exchange for their powers during battle. They can be used to distract opponents, to siphon their chakra, to restrain them, and more.
Most characters in My Hero Academia have a superpower called a quirk. Kouji Kouda's quirk, Anivoice, allows him to communicate freely with animals, as well as control their behavior. Not only does this allow him to do things like sending swarms of insects or hordes of deer after his enemies, but it also gives him someone to talk to when he can't handle humans due to social anxiety.
As you might have guessed from her name and her spiderweb eyes, Arachne Gorgon boasts control over spiders. She can not only utilize many spider-based techniques, including the terrifying Madness Wavelength, which allows her to drive her opponents insane and make them feel like they're being torn apart by spiders. She can also physically restrain and paralyze people using spiders' webs and threads. But it's not just that Arachne has spider-like abilities of her own - she can also generate spiders that are separate from her body to do her bidding.
This trope applies to just about everyone in the Pokémon universe since most of the human characters spend their time training animal-like creatures for combat. Of course, most of them are able to do things that real-world animals can't - there's no mouse/rabbit hybrid on earth that can electrocute people. Ash Ketchum is the main character of the show, and probably the most dedicated to being a Pokémon master, so he gets the spotlight here.