Harry and company don’t steal Fluffy from Hogwarts, nor did they bring him there. Hagrid did, after he bought him from a “Greek chappie” under less-than-legal circumstances (*cough* animal trafficking *cough*). This is a far cry from the violent episode you’re probably imaging with Heracles literally dragging Cerberus out of the Underworld. But Timothy Gantz points out in his 1993 book Early Greek Myththat the iconography of this episode, and its development, depicts a shift in attitude about Heracles’ involvement in this labor. Early vases show him violently storming the Underworld to steal Cerberus; later ones show a more gentle encounter with Persephone allowing Heracles to borrow Cerberus. So while Heracles doesn’t necessarily steal Cerberus every time this episode is recounted, he does participate in the movement of an exotic animal across borders — with no real intention of returning him.
But we shouldn’t pillory Heracles as a proto-animal trafficker. Animal trafficking as we know it, the illicit trade for everything from bushmeat to elephant tusks to illegal house pets, just didn’t exist in the ancient world. What did exist in Antiquity was animal hunting, specifically exotic animal hunting. Think crocodiles from Egypt, panthers from Greece, and elephants from north Africa. Big game from conquered locales played a huge role in the Roman Empire, marching alongside war captives during the imperial triumph. Our earliest account of foreign animals’ participation in processions comes from Josephus’s description of Vespasian’s and Titus’ triumph where “beasts of many species were led along, all decked out with the appropriate adornments” (7.136). Exotic animals, dressed up and subjugated like their human counterparts, were paraded through the streets as prisoners, then later slaughtered in the amphitheater during the ludi, or games. The more fantastic the beast, the better.
So while these fantastic beasts weren’t trafficked, they were displayed as symbols of Rome’s power. And that’s what Cerberus/Fluffy does for his conqueror. Like the emperors Titus and Vespasian, in capturing and later displaying Cerberus to Eurystheus, Heracles didn’t display the might and prowess of the hellhound but his own. Similarly Rowling displays her own literary and cultural prowess when she places Fluffy at the trapdoor leading to the Sorcerer’s Stone. Here, she takes a prominent, seminal image from Western literature and adapts it for her own purposes. And she does so in a pretty clever way by placing Fluffy at the beginning of Harry’s katabasis (a descent into the Underworld) and associating him with heroes like Orpheus, Heracles, and Aeneas.
But who owns Fluffy? The ancient Greeks are no longer alive, and their stories no longer hold any real religious weight. You could point fingers at Rome, but Rome “appropriated” Greek culture long ago (part of a larger drive in Roman society to assimilate conquered cultures). Greek mythology now resides in the foundations of Western literature and art: you can plagiarize its particular iterations but not its general ideas. This is why Greece can argue for the repatriation of the Parthenon Marbles but not, say, the dismantling of Washington, D.C. And yes, on Cerberus you could argue that Vergil cribbed Homer, and Dante Vergil, but whoever came up with Cerberus first is long gone — and he’s ripe for the taking.