The discussion when it comes to animal companionship in the Harry Potter series often centers around owls, cats, and toads. If the discussion evolves beyond this, questions and comparisons between a person Patronus and their Animagus form come up.
Here we are going to look beyond the wizarding world into the world of Philip Pullman's, His Dark Materials. What daemon might each Harry Potter character possess? Would a daemon match a Patronus or an Animagus? Or would it be a different representation all together? We are going to go with different, and you can look at some of our reasoning below.
James
James Potter, the Potter patriarch, father to The Boy Who Lived himself, had a very complicated backstory once it all came into the open. While Harry's father was very brave, in defending his family against Lord Voldemort, he was a bit of a showoff and a bully as a child and teen. James grew into a decent human being, which is why he certainly wouldn't have been saddled with something like a dung beetle. James Potter's daemon would likely have been a collie. Collie's are known for bravery (think Lassie) and are also smart, good with children, eager to please, and good at agility games. With James' history as a father and Quidditch player, it certainly fits.
Lily
Lily Potter had a less complicated backstory than her husband, always known for her kindness and generosity. She was also known for her red hair and her incredible sacrifice for her only son, Harry. Lily's daemon would have been an orangutan, and not just because of the red hair.
Orangutans are extremely strong and can be dangerous when they have to be, They are also one of the best parents in the animal kingdom, protecting their young for many years and, doing such a good job, that children continue to return to visit throughout their lives.
Harry
Harry Potter, The Boy Who Lived, is best known for his heroic self-sacrifice (on more than one occasion). While in the books Harry's Patronus takes the shape of a stag, his daemon would more likely have been a gray wolf. Gray wolves are known for certain self-sacrifice, bringing back food to the pack, helping the old, sick, and injured. They must also care for the young, who are born blind and deaf until almost a year old. Wolves are also loyal, mating for life. And, as a boy who grew up abused by his aunt and uncle, a wolf at his side would likely have been quite a comfort to Harry.
Ron
Ron, the youngest of six brothers and best friend to the Boy Who Lived, would likely have hoped for a daemon that stood out, like a woolly mammoth or a Bengal Tager. Odds are Ron would have been better matched with a labrador retriever. Friendly, outgoing, and affectionate, like Ron, a labrador would have had no trouble welcoming Harry on the Hogwarts Express. Eager to please and incredibly loyal, there's no doubt that Ron and the lab have quite a bit in common. An enthusiastic athlete, a lab would have no trouble keeping up with Ron and his brothers in the backyard at Quidditch practice.
Hermione
Hermione, "the brightest witch of her age," would certainly have needed a daemon who could keep up with her academic rigor, which is why a chimp likely would have been the final form of Hermione Granger's daemon. Chimps know how to use tools, are excellent climbers (imagine the books on the highest shelf of the Hogwarts library) and are also excellent communicators. It would make a lot of sense for Hermione to pair with a chimp in the world of His Dark Materials.
Neville
Neville Longbottom, the reluctant Gryffindor, could have used a daemon with a bit more confidence than he had at the beginning of his Hogwarts career. Neville may have had to grow into it, but his daemon would have been a German Shepard. One of the bravest dog breeds around today, the daemon would have made it clear to everyone that Neville belonged in Gryffindor, even if the boy himself doubted it. Loyal, confident, and easy to train, the daemon would have looked like the opposite of his human (except where loyalty was concerned), that is until the whole world watched the pair together face down Lord Voldemort.
Luna
Luna Lovegood, a girl who always marched to the beat of her own drum, would have been followed around the Hogwarts grounds by a Tea Cup Pig. Perhaps people might have made fun of her for having a pig as a daemon but that's only because people don't understand pigs, the same way most didn't understand Luna. Pigs are incredibly intelligent and also incredibly clean. They only get a bad reputation because, without sweat glands, they need to roll around in the mud to stay cool. Luna and her tiny pink sidekick would have been perfect for one another.
Ginny
Ginny Weasley, unlike her brother, would not have had a dog, but rather a crow as her daemon. Aside from Ginny's excellent flying abilities (she did go on to be a professional Quidditch player after Hogwarts) but also because crows are very clever. Ginny had to sneak brooms away from her brothers to practice flying. She was able to balance a full social life and top grades. It's not hard to see Ginny sitting, crow on her shoulder, trading knowing looks with her daemon about anything and everything.
Dumbledore
Despite his Patronus, Dumbledore's daemon would not be a phoenix (even if it wasn't a magical creature, to begin with). Dumbledore's daemon would instead be a Cuckoo Bee. While Dumbledore itself translates to Bumblebee, the Cuckoo Bee matches far more of Albus' personality traits. Cuckoo bees are known for hiding their offspring in the hives of other bees and have them take over the child care (sound like anyone else who was left on a doorstep?). They know what they want and they will do what they have to in order to succeed.
Voldemort
The one character who offers us no surprises is the Dark Lord himself, Voldemort. There is no question Voldemort's daemon would be a snake.
Aside from his being a descendant of the parseltongue prince himself, Salazar Slytherin, Voldemort also already showed the world a literal manifestation of his soul in animal form, when he made Nagini into one of his Horcruxes.