"Why would a human say “I want to fight like that animal?” "
The Human sees something useful in the fighting of That animal. If a person had to fight either a Human or an animal~ several at least might prefer to fight a Human. If you can reason why they would Not prefer to fight an Animal you then can see why a Human might would-like to “fight like that animal.”
“Animals don’t try to copy the movements of other animals.”
But they might learn them to counter them or use the nature of the creature to it’s advantage. If I learn that a snake strikes with its head, I can stay away from that end. Also, they only strike within certain ranges. Helpful notes perhaps for best outcome interaction.
"A bear isn’t going to go “I want to fight like a tiger.” "
…because they fight similarly already with similar skill levels. Speed, Power, claws, teeth, vocal startling at least.
“…So why would a human assume that the best style of fighting for a human would be to copy the movements of a non-human?”
Because for the eons ago, about which is a starting point for fighting like animals, animals were dominant one on one fighters against humans perhaps.
Mimic the winners. Humans mimicked animals in perhaps several ways that made the mimicking beneficial. They mimicked tooth/fang and nail/claw/talon in using edged and pointed tools eventually making knives–still a useful mimicking of animals today.
Speed and power were not across the board readily matchable; treachery was used to counter skill. Even today learn how the system works, learn the nature of the target..perhaps not copy the animal, yet learn the animal/target for effective tricking, trapping, deceiving…
Striking first and from a distance (pouncing/striking (of snakes) was translated into long reach (poles, throwing, launching weapons (slings, bows (catapults), guns (and their ammo)).
Mimic animals foraging. One of the better ways to without risk find out what to eat (besides each other)~.
Mimick animal stealth for hiding hunting. Note: Humans instinctive skill is run/hide. Human children the world over will run and hide as game before fighting. ~ Perhaps, if so.