The Bible is full of animal analogies. Lions, doves, sheep and cattle are especially prominent: Jesus himself was regularly compared to lambs, lions and, in later Christianity, unicorns.
The Iliad is heavy with animal similes when describing characters' emotions and actions.
Many of the noble houses in A Song of Ice and Fire have animals on their coats of arms, and are commonly referred to by these animals as well as showing an awareness of how they are 'supposed' to act to fit the animal characteristics; the series has a lot of fun with the possible symbolism. For example, the grim, grey, and cold-dwelling Starks have a direwolf on their arms, while the blonde, vain, and aggressive Lannisters have a lion. Littlefinger deliberately chooses the small, bright, and harmlesscunning thief, the jaybird, for his arms to appear nonthreatening. Crows seem to act as the motif for the series as a whole, symbolizing war, ill omen, and death.
To top it off many characters are refered to by animal nicknames based off their traits. Like Sansa Stark is called "little bird" (okay maybe by one guy) for her ability to repeat "Pretty words". Also, Sandor is often called dog or The Hound.
The Starks, Targaryens, and Dothraki have a closer relationship with their animal sigils (direwolf, dragon, and horse, respectively) than most (not like that). The Stark children all get direwolf companions early in the first book. The Targaryens rode dragons to war, and some of them seem to have a resistance to heat. The Dothraki, of course, consider horses essential to their nomadic lifestyle.
The Neanderthals depicted in Jean Auel's Clan of the Cave Bear have individual animal "totems"; the heroine, Ayla, has a cave lion, a controversially strong totem for a woman.
Literally used in His Dark Materials where your daemon's form (an animal) represents yourself and is part of your soul. Since children have not yet truly defined who they are in life, their daemons can shape shift until they mature.
The Song of The Lioness book series features a female knight called Alanna, called "The Lioness" thanks to her courage and tenacity.
There are the protective spirits brought forth by the Patronus Charm in Harry Potter—Harry's Patronus is a stag, which is the animal his father could turn into, Ron Weasley has a terrier, and dotty but quick-thinking Luna Lovegood has a hare. The Patronus of Severus Snape turns out to have a critical symbolic significance recognized even by characters in the scene. Dolores Umbridge, who fills her office with adorable kittens and has a cat Patronus, presents an atypical symbolic association for felines; the cute kittens and sleek cat Patronus (which we see only in a scene where it magically shields Umbridge against the consequences of her own ongoing miscarriage of justice) represent cloying, false sweetness, politeness masking lethal intention, and near-pathological self-justification.
Ginny's Patronus is a horse, Mr. Weasley's is a weasel (unusual for a "good" character), Hermione's is a playful otter (J.K. Rowling's favorite animal), Irish Seamus Finnegan's is a fox, stolid Ernie Macmillan's is a boar, pretty Cho Chang's is a swan. All the Animagi in the series (wizards who can turn into animals) seem to have the same animal form as their Patronus.
Minerva McGonagall has a cat Patronus and transforms into a cat, so it's likely that there are alternate ways to interpret the Patronuses based on the witch or wizard's personality.
Apparently the Patronuses can also change forms if the witch or wizard's personality changes, or to reflect the person they love. Lily Potter, for example, had a doe to match her husband's stag as did Severus Snape, who loved her, and Tonks's changed into a werewolf after she fell in love with Lupin.
Animagi can also serve as a character motif. James was a stag, representing a strong animal which is wild but also overly prideful. Sirius is a dog; something clever and also devotedly loyal. Peter was a rat, specifically Ron's pet rat scabbers. This could be interpreted positively as being dexterous and clever, but also as being mistrustful and well...rat-like.
Also, for all that Snape's patronus is a doe in honour of Lily, he is often associated with bats, something which is dark and 'evil' but also misunderstood and associated with witchcraft.
Entire groups of people can also take on the animal motif of their house, but to a lesser extent. Gryffindor has a lion, showing bravery. Hufflepuff a badger, signifying plain-ness, but also hard workers. Ravenclaw has an eagle, for intelligence and knowledge. Snakes are the symbol of Slytherin, and associated with evil, but also traits such as intelligence and ambition.
The Animorphs all have signature animals they often turn into, particularly in battle. For example, Marco, the team clown, typically transforms into a gorilla, which is big, bulky, but still capable of delicate manipulations with opposable thumbs; The Hero, Jake, is a tiger; Blood Knight Rachel is a grizzly bear; Tobias is a hawk (self-sufficient and needs lots of space); Team Mom Cassie is a wolf.
The Grapes of Wrath has in its third chapter a tortoise trying to cross a road and getting run over by a truck. In the fourth chapter, we meet Jim Casey, who's described as having a "long head" and a "beaked" nose.
Jorge Luis Borges writes a tiger somewhere into many of his stories.
Songs of Innocence and of Experience employs the recurring imagery of different animals to represent innocence, e.g. "The Lamb", and experience, e.g. "The Tyger". In "THE FLY", the narrator compares himself directly to a fly with reference to human mortality and vulnerability.
An in-universe example: In Dragaera, each of the 17 Great Houses is named after an animal. However, since a lot of those animals aren't from Earth, the similarities between the personality traits of the House and the traits of the animal are much better understood by the characters than by us.
In Stardust, a character remarked when Tristran is transformed into a dormouse by a witch, that she isn't sure whether the spell simply transformed Tristran into a dormouse or it had chosen the form that Tristran is most like.
From Western Animation//Coraline, the Other Mother is especially given imagery of a spider. Particularly her real form and not to mention how her kind is known to eat their offspring.
Bugs are everywhere in the film as well. The Other Mother added them to the Other World because she thought Coraline liked bugs.
Each chapter of The Darksword Trilogy starts with a small, symbolic illustration. Joram is represented as a frog wearing a crown, referencing his Royal Blood; Simkin is represented as a crow wearing a jester's cap, referencing his Trickster Archetype nature and status as The Fool.
The Hunger Games has President Snow identifying with a snake. Katniss has been compared to a mockingjay.
In Death series: Roarke has been compared to a wolf in Ceremony In Death, Vengeance In Death, and Promises In Death. Promises In Death actually had Eve and several women discussing the men in their lives and comparing them to animals. Nadine got a turtle of a man. Trina got a snake of a man. Mira got an owl of a man. Peabody got a puppy of a man. Mavis got a bear of a man. Louise got a cheetah of a man. Nadine theorizes that Roarke is a panther, but Eve ends up saying "Wolf," and Mira points out that when wolves mate, it's for life.
In the Belgariad, each of the gods has a totemic animal: Belar's is a bear, Torak's is a dragon, Chaldan's is a bull, Issa's is a snake, Nedra's is a lion, Mara's is a bat, and Aldur's is an owl. Each god, and his chosen race of people, take on the personality characteristics of their animal, and in some cases go beyond: Barak turns into a bear when a certain condition is met; Salmissra is transformed into a serpent as punishment for her particular meddling; Torak's successor transforms into a dragon on occasion in an effort to impede the party; and the female sorcerers under Aldur's tutelage prefer the form of the owl for themselves (the males prefer the wolf). The sorcerers also wear an amulet each bearing the image of their chosen form.]
