Interspecies Adoption

Even in settings where Talking Animal, Civilized Animal, or Funny Animal animals exist, adoptions will still happen between parents and children. Some parents are caring enough to take in a child, regardless of their species. They may be animals raising a human child, humans raising an animal child, animals raising animals of a different species, or any combination of the three. It doesn't matter, as adoption is still adoption, and parents will always be parents. The subject of the adoption may or may not even be an issue among society.

This is a supertrope of Raised by Wolves and Raised by Humans, as well as cases that aren't covered by either trope. It may also overlap with Moses in the Bulrushes and (especially in comic settings) Oblivious Adoption.

Examples of Interspecies Adoption include:

Anime and Manga

In Transformers Victory, Star Saber, the new leader of the Autobots, adopts a human boy named Jan, who lost his parents to a Decepticon attack.
In You Are Umasou, a Maiasaura raises a Tyrannosaurus from birth, who in turn, after running away from home, adopts a baby Ankylosaurus after being mistaken for its father.

Comic Books

Hellboy is a demon who was adopted by a human parent named Dr. Bruttenholm and raised under the watch of the Bureau of Paranormal Research and Defense. He was raised to appreciate humanity, even if he constantly has to deal with the fact that he's frighteningly different from them anyway.
Superman is probably the most famous example of this trope. He's an alien sent to Earth as a baby, and he grew up on a farm by the Kent family.
Teenage Mutant Ninja Turtles: four adorable baby turtles fall into a sewer and are adopted by an ordinary rat (other than the whole "knows ninjutsu" thing). Only later do the five mutate and become intelligent.

Fan Works

Undocumented Features has Chad Collier, a Kilrathi raised by humans, and Marty and Eiko Rose take in and raise Wakko, Yakko and Dot ~~Warner~~ Wharn'yr.

Film

The Stuart Little movie did this to remove the Fridge Logic from the original book (where the titular character just had human parents for no apparent reason).
Giant panda Po's father in Kung Fu Panda is a duck; this goes unmentioned in the first movie, but becomes a minor plot element in the sequel.
In Thor, Loki turns out to be an Ice Giant adopted by Odin and raised as his own under a glamor.
Elf: Buddy is a human adopted by Santa Claus, and raised among elves.
Disney's The Ugly Dachshund. A Great Dane thinks he's a dachshund because he was raised by a dachshund mother.
Aladar from Dinosaur is an Iguanodon raised by lemurs.
Babe the pig is adopted by Fly who is a dog.
Most of 'Leafie, A Hen into the Wild' revolves around a hen and her Happily Adopted son (a duck).

Literature

In the Discworld novel Guards! Guards!, Carrot is a human adopted by dwarves. He is completely oblivious to this, even when his adoptive father tries to explain that there's a reason he was always too tall to fit in dwarf passages correctly. Notable in that even after it's been explained to him and he's accepted that he is biologically a human, Carrot still considers himself a dwarf, albeit a very tall one.
"The Ugly Duckling" is a classic example of this trope. A swan raised by a duck.
RuneScape: Betrayal at Falador, Kara was adopted and raised by dwarfs.
The children's book Tyranosaurus Drip by Julia Donaldson is about a duckbill dinosaur "adopted" by tyranosaurs. (Mother Tyranosaur can't count, and doesn't realise there's one more egg than there should be.)
The Sheep-Pig by Dick King-Smith, the book Babe was based on, likewise has Babe raised by Fly the sheepdog.

Live-Action TV

Worf was adopted by the Rozhenko family, a human couple from Russia. In turn, Worf's son Alexander Rozhenko was also raised by Worf's adoptive parents.

Video Games

Mallow from Super Mario RPG, originally from Nimbus Land, was adopted by Frogfucius in Frog Pond, and grew up largely believing he was a frog "who can't jump".
Adell in Disgaea 2: Cursed Memories is a demon adopted by humans as a result of his biological parents leaving to fight fake Zenon. They fail.
Penn in Soul Nomad and The World Eaters is adopted by the Nereids. This is part of a breeding plan when he turns 18.
F-Zero's Leon, depending on the game (although both examples still fit). In X, he's adopted by Mrs. Arrow and her husband Super Arrow, a crime-fighting Superhero couple. According to GX, however, he was instead adopted by Fable, a soldier belonging to the race that attacked Leon's homeworld twelve years ago (which led to the poor kid losing his parents and his left eye, as well as Zou becoming an economical dump) who rebelled when he became disgusted by their tactics. Leon's a bipedal cat-like alien, the Arrows are human, and Fable is another species of extraterrestrial being.
After the Time Skip in Final Fantasy VI, the half-human, half-Esper Terra is shown to have adopted some children from Mobliz whose parents were killed by Kefka's Light of Judgment.
At the end of Final Fantasy IV: The After Years, the last remaining Maenad is adopted by Rydia and christened Cuore. This is a bit of an ambiguous example as the Maenads are a race of Artificial Humans made in Rydia's image by The Creator.
After her parents were killed by Space Pirates, Samus Aran was adopted by the Chozo, an ancient race of bird-like creatures that raised her and trained her to be what she is.
In Devil May Cry 2, a big deal is made out of Lucia being a flawed demonic construct of Arius instead of Matier's biological daughter. After going through a Heroic BSOD and struggling with the ramifications of her true nature for a bit, Matier comforts Lucia and outright tells her, "You are my daughter." This gives Lucia the strength to carry on and break out of her Cloning Blues complex.
Neverwinter Nights 2. The protagonist's race can change, but the foster father is always the same.

Web Comics

Lindesfarne in Kevin and Kell: Kevin (who is a rabbit) and his (later-ex-)wife adopted Lindesfarne (who is a hedgehog) before Kevin met Kell.
The trolls in Homestuck. It's societal norm for them to be raised by another species. However, these guardians may or may not be totally sentient, and not all of them are good parents, so it also delves a bit into Raised by Wolves.
Selkie is about an amphibious young girl, and the man who adopts her.
Buwaro, Sakido and Iratu, all demons, are adopted by the angel Darius in Slightly Damned.

Web Original

Quite a few characters from Cerberus Daily News. Thus includes (but is not limited to) a Turian who was adopted by a human couple (Gahars Patnus), a human who was adopted by an Asari couple (Some_Random_Merc/Johnny), a Drell who has recently adopted a human infant (Rohim), a human whose legal guardian and father-figure is a Turian (Wildflower/Flower and Davril, respectively), and a human who was raised by a Quarian (Human Quarian/Kolya).

Western Animation

Heffer was raised by wolves. The Wolfe family to be specific. They were just going to eat him, and started fattening him up, but grew to love him and raised him as their own. The "birthmark" on his rump is where the wolves were going to divide him up.
SpongeBob and Patrick adopted a baby scallop in one episode. It left other citizens quite confused when they thought about the biology involved.
Dinosaur Train: Buddy the T-Rex is adopted by a pteranodon family.
CatDog continually brought up one particular issue over the course of the series: Where did CatDog actually come from? For as long as they can remember, they'd always lived on their own. The series ended with CatDog going on a long journey to find their parents. It was never revealed how CatDog came to be, but for a brief time after they were born, they had been raised by a frog for a father and a sasquatch for a mother. They were all separated in a storm.
Darwin from The Amazing World of Gumball used to just be the Waterson family's pet fish, but he developed intelligence, grew legs, became Gumball's best friend, and the family adopted him as one of their own.
In My Little Pony: Friendship Is Magic, Twilight Sparkle (a unicorn) had to induce a dragon egg to hatch as part of her entrance exam in magic school. She ends up being a Cool Big Sis/Parental Substitute to him, who serves as her underling and assistant.
Buzz Lightyear of Star Command probably topped this concept beyond most other examples by having a human girl being adopted by two robots.

Other Media

This Bad Habits strip, which is also an Oblivious Adoption.
There is a famous joke about a turtle which constantly climbs up a tree and jumps down with its legs spread. After a few attempts (and the turtle getting quite a few traumas), a bird watching it from nearby asks its mate, "Should we tell our son he's adopted?"

Real Life

Real Life examples involving animals like dogs, cats, and rabbits are a mainstay of websites like Cute Overload.
One of the strangest incidents of this kind occurred in Samburu, Kenya: a lioness adopted an oryx.