Fullmetal Alchemist had a variation on this. Scar, the neighborhood serial killer, comes out to violently attack our heroes. May Chang attacks the Elrics when they try for a finishing blow, and runs off with an injured Scar.
One of the first stories in Gintama has the daughter of an alien diplomat trying to rescue her unknown massive alien being. Gin cuts it up.
Inverted later when Shinpachi thinks Kagura's new alien giant dog is playing with her, Gin informs him that the dog's attacking her and she's too powerful to notice.
Bleach introduces Nel in this fashion. Ichigo "rescues" her from a group of arrancar who turn out to be her friends.
Incidentally, Nel is playing "endless tag" with her friends, and since "Nel is a masochist" (and a loli with her arrancar mask covered), their "game" looked an awful lot like three hollows chasing a sobbing little girl. Actually, that's exactly what it was.
In Digimon Tamers, this probably would have happened with Antylamon (the last Deva yet to be revealed) had Susie not been given her Digivice before Takato and Henry could think of attacking. And Zhuqiaomon reverts her to Lopmon shortly afterwards, confirming (see Redemption Demotion) her Heel-Face Turn.
This is featured multiple times in A Series of Unfortunate Events, both the movie and the books, with the Incredibly Deadly Viper — most notably when it "attacks" Sunny.
Harry Potter: While Hagrid is far from a cute young girl, his attachment to his exotic pets fits. He names the three-headed dog Fluffy, he is upset when he has to send his baby dragon Norbert away, and he cries buckets when the giant spider (Acromantula) Aragog dies. He was expelled from Hogwarts for keeping dangerous critters as pets, including Aragog. Harry and Ron are often at a loss to understand him, though the Trio does recognize why he is upset over the scheduled execution of an innocent hippogriff.
Hagrid himself is a version of this. Giant hairy scary guy who can bend rifle barrels with one hand comes knocking down the door? If the Dursleys were proper foster parents instead of assholes who didn't give a crap about Harry, they'd probably freak out when Hagrid approached Harry. Interestingly enough, he got the reverse treatment in the wizarding world. When Hagrid was outed as a half-giant, despite the attempts to drum up the sensationalism, Dumbledore was bombarded with a barrage of letters telling him not to dare sack Hagrid because he was one of the best men the parents knew and they trusted him to keep their kids safe.
Almost happens in Earth's Children, when the lion Ayla raised from a cub comes back to visit her in the human village she has moved to. She manages to stop the hunters before they harm the lion.
The Enchanted Forest Chronicles contain multiple examples, especially in the first book, Dealing With Dragons. A princess, Cimorene, runs away and volunteers to be the captive of a dragon; she then spends her time trying to stop knights from fighting the dragon because she doesn't want them to rescue her.
In Something MYTH Inc., Gleep the baby dragon gets accidentally shot because a trigger-happy bowman thinks he's attacking when he just wants to play.
In Keeper of the Isis Light, the main character's gigantic pet is killed when it tries to engage a human in a game of fetch.
Early in Tanith Lee's The Dragon Hoard, while Jasleth is looking for a quest to join or a monster to slay, he comes across a prince tussling with a two-headed monster, but it turns out that the monster is the prince's pet, they were just playing, and neither appreciates Jasleth's offer of assistance.
Voltaire's Candide published in 1759 had this in the story. The heroes rescue some women running from apes and it turns out the apes were their husbands.
In the Doctor Who story The Rescue, the relationship between 20th century companion Barbara and 25th century companion Vicki starts off badly when Barbara "rescues" Vicki from being "attacked" by a 6-foot-long, tooth-baring (vegetarian) lobster-monster... by shooting it in the face with a flare gun.
A sketch on The Dave Allen Show had a slavering werewolf doing a typical horror movie stalk of a Distressed Damsel. When he finally cornered her, she holds up a leash and calls "Walkies!" while the werewolf bounds about like an excited puppy.
A pregnant woman was being followed by an armor-plated demon. Angel kills the demon but discovers that it was protecting her from other demons.
Cordelia, trapped in a unknown dimension ends up being chased by a vicious beast who catches her and... licks her face?. Whew. Then the pet's owner showed up, congratulated the "dog" and chained Cordelia. Unfortunately, it turns out that while the beast was a pet, it was a hound hunting... humans.
The June 2009 trailer for Team Ico'sThe Last Guardian sets this up.
Averted in Shadow Hearts: Covenant. A giant cat attacks you at one point. This is Lucia and Carla's pet, Andre... but they don't mind you beating up their cat because, frankly, the cat didn't have a reason to hurt you and you were just defending yourself.
Becomes a major character-driving plot point for one villain in Tales of the Abyss, where the monster in question was the adoptive mother of the little girl, who naturally spends the game trying to avenge her giant carnivorous mommy.
The prologue of Vindictus, where Tieve is trying to persuade the mercenaries to let her reason with her friend the giant fuzzy spider before they start shooting.
In the Pilot Movie of Foster's Home for Imaginary Friends, Eduardo saves Mac (whom he had followed around, since he's too shy to introduce himself) from an Extremesaur. Bloo thinks Ed is attacking Mac and jumps him. Eduardo would have easily dispatched Bloo if he wasn't such a scaredy-cat...
An episode of Avatar: The Last Airbender had Aang trying to find a crazy king's lost pet Flopsie as one of three deadly challenges. He thinks it's the cute and cuddly rabbit being chased by the 7 foot tall gorilla-goat thing. He eventually figures out the trick and even learns an Aesop about it (after the climatic fight with the king, of course).