A Mature Animal Story is a work designed for and marketed toward adults and older teens featuring anthropomorphized animals as its main cast, and which contains content that is generally considered inappropriate for little kids.
Traditionally, cartoons, comics and other stories featuring animals as major characters are treated as light, whimsical entertainment suitable for small children. While adults may enjoy such works, they are primarily marketed toward young kids, and are careful to avoid material that is either too explicit or too serious and sophisticated for children to understand. While such works still might contain some conflict, or even a Disney Death, it is always at a level considered "Family Friendly" and "Safe for Kids".
Also, until recent advances in CGI, anthropomorphized animals in visual mediums were mostly confined to rather silly looking costumes, cartoons and comics. These features were generally associated with less serious children's stories, and therefore weren't going to be all that big on more "mature" topics like violence. Talking Animals in particular are often considered kids stuff.
A Mature Animal Story breaks away from this perception. The primary audience is adults and/or older teens. Stories still might be comedic, but they will also often have a serious and dark side to them. Many such works are almost entirely drama instead of comedy. They often contain graphic violence, sexuality, and profanity.
These stories involve animal characters ranging from Largely Normal Animal to Petting Zoo Person on the Sliding Scale of Anthropomorphism. They frequently take place in a World of Funny Animals.
This is a Sub-Trope to Animation Age Ghetto and What Do You Mean, It's Not for Kids?. Compare Xenofiction.
Please note that if it is marketed as a work for children, it is not this trope.
Similarly, Penguin Memories is a serious drama in which a Vietnam vet deals with his inner demons—and yes, all the characters are penguins.
Ginga Nagareboshi Gin and its sequel, Ginga Densetsu Weed are about talking dogs. Kid-friendly, right? Hardly — there's some very disturbing stuff (including a dog who castrates his enemies) in Weed, and Gin is even more violent, and that's not even counting the manga.
Oruchuban Ebichu is a comedy about a cute little hamster and her unmarried owner. Almost every joke is about sex.
Cat Soup is an abstract, existential short film based on the extremely dark comics of artist Nekojiru. But hey, all the characters are cats!
Usagi Yojimbo involves adorable anthropomorphic animals, the main character being a rabbit. The setting, however, is Feudal Japan with all of the cultural baggage thereof, including kirisute, the traditional right of the samurai to kill any commoner who offends him. There is exploration of death, relationships (including forbidden relationships, illegitimate offspring, widowhood, etc.), various extremely nasty monsters from folklore, political intrigue, crime and punishment, and a very messed up judicial system.
Maus is a retelling of the author's father's experiences during the Holocaust, including his stay at Auschwitz. Jews are mice, Germans are cats, Americans are dogs, Poles are pigs, French are frogs, Swedes are reindeer, and Gypsies are moths.
Benoit Sokal's Inspector Canardo is about a detective duck in a world of talking animals. The protagonist is a drunk, people get killed, prostitutes and rapists are commonplace. Definitely not for children.
Swedish comic strips Arne Anka and Rocky are filled to the brim with funny animals. They're also filled with alcohol consumption, sex and deep, deep cynicism.
We3, by Grant Morrison, is about three talking animals trying to find their way home; the covers feature "missing pets" notices written in childlike style. Kids'll love it, right? Sure! Except for the scenes featuring the cybernetic animal soldiers literally tearing apart the soldiers meant to come kill them, the part where the rabbit explodes while hurling itself at a car, and all sorts of graphic violence in between. Oh, and it's being adapted into a movie directed by the guy who made Kung Fu Panda.
Blacksad is a Film Noir style series of graphic novels about a black cat private detective named John Blacksad and his various cases set in a World of Funny Animals version of 1950's America. It doesn't hold back from showing blood, fatal injuries, wounds, controversial topics (e.g. racism), and sex.
Grandville - the main character is an anthropomorphic badger who is a private investigator. It's set in a Steam Punk fantasy setting, and features plenty of sex, violence, and politics.
The characters in the Sandbox comics by Swedish artist Joakim Pirinen are antropomorphic teddy bears, while their plots often deal with sex, drugs, violence, and disturbingly psychedelic visions.
The infamous Air Pirates Funnies was a short-lived underground parody comic which featured Disney characters in adult situations, such as having sex and taking drugs. Disney sued the makers of the comic for copyright infringement, which apparently was exactly what they wanted.
The Unfunnies is this trope all over: Hanna-Barbera-style characters start talking about pedo-pornography as soon as the second page, and it goes further downhill from there. Of course, the Hanna-Barbera element also includes the occasional cartoony humans thrown in for good measure, but the comic still counts.
The Pride of Baghdad has blood and at one point a lioness flashbacks to being gang raped (It's not shown, but still.) Also, they all die at the end.
Films — Animated
Watership Down. The introductory animation has animals being killed, and this continues throughout the movie. At one point Bigwig is trapped by a snare around his neck and is clearly suffering. The original warren the protagonists come from is a moderately harsh dictatorship, and Efrafa (General Woundwort's warren) is much worse. There's also a lot of blood and at one point rabbits are buried alive.
Hazel: Don't make it too grim, Holly.
Holly: "Grim"? I haven't even begun.
Felidae. To quote the entry: "Despite being an animated film, it is definitely not for kids. The film contains graphic violence, disturbing images, adult language, and cat sex. Yes, cat sex."
The Plague Dogs is an animated film based on a book by Richard Adams, author of Watership Down. It's about two dogs that escape from a research lab and attempt to live in the wild. It starts with a dog drowning as part of an experiment involving how long it can tread water. The tale highlights the cruelty of animal research and vivisection. In America, the film was rated PG-13.
Anna Sewell wrote Black Beauty as a lengthy social commentary on animal welfare, politics, religion, and social justice, including living conditions for the working-class poor. While there's probably not anything that's inappropriate for younger readers (aside from a couple rather gruesome depictions of dead and suffering horses), much of it would probably go straight over their heads. Versions of the book are frequently marketed to children, but these have typically been heavily edited or rewritten entirely.
Animal Farm is more or less a breakdown of Josef Stalin's part in the Russian Revolution, portrayed via animals throwing a farmer out of a farm and then running it themselves. The pigs are the communists, they raise dogs as their police, and the sheep pledge their full support to everything said to them without thinking about any of it. All sorts of Fridge Horror ensues, such as the horse being sold to a glue farm while the pigs periodically read letters they claim the horse wrote to them from his retirement pasture. Fortunately, nobody really thinks of it as a book for children, although it is often taught at middle school level (13/14-year-olds can usually grasp what's going on, and if you teach it alongside the actual history of the Russian Revolution and Stalin's regime, it makes for a good introduction to the concepts of allegory and satire).
Lackadaisy is a webcomic about a bootlegging gang in Prohibition-era St. Louis, with plenty of gunfights, killings, and several sociopathic characters. There is painstaking attention to period-appropriate detail, with one exception: all the characters are anthropomorphic cats.
Jack is a Furry Webcomic about a rabbit who is also The Grim Reaper. Most of the stories take place in the afterlife or involve death in some form. This comic can get veryNSFW at times.
Drugs And Kisses features the misadventures of several pot-smoking, vodka-swigging anthropomorphic animals.
Tasakeru is a series about cute, fluffy talking animals like squirrels and rabbits... which involves bloody warfare, racism (speciesism?), religious intolerance, Mind Rape, and not a little innuendo.
The Cat Piano may qualify, being about Petting Zoo People / Anthropomorphized cats who were being kidnapped in order for the titular piano to be played (it stabs a nail through a cat's tail every time a key is struck, making the cat yowl the note). Particularly the ending, where the catnapping human slips in his own blood and falls off a tower to his death.