Just because humans are uncomfortable with the subject of nudity, doesn't mean we're the only species to get our nudity censored in fiction — it is just as common for regular everyday animals to get this treatment as well.
This is usually a concession of realism for the sake of the artist, as most artists would prefer to avoid drawing an animal's genitals or anus themselves, taboo or no taboo. It can also be a product of stylization, to which animals are especially prone, and which tends to remove unappealing or unimportant features, oftentimes to make them cuter (for example, stylized dogs tend to lack claws and lips). Othertimes (by which we mean most of the time), whether or not Fido has gonads is way beside the point so it's simpler to leave them out unless you mean to draw attention for a narrative purpose or a cheap gag.
This is technically Truth in Television for animals such as birds, reptiles, amphibians, fish, and invertebrates, whose genitalia are typically visible only as a small underside 'vent' between the rear limbs (and easy to miss even without feathers or scales to obscure them). That's not a form of censorship, it's just Law of Conservation of Detail (remember: Nobody Poops).
On the other hand, when depicting mammals (save for platypusesnote Platypi? Platypeese? and echidnas), there's a common assumption that an animal's natural fur will adequately preserve the creature's modesty. However, observation of most real mammals will show that their fur is actually thinner in those places (so it doesn't get soiled when nature calls, and is easier to clean if it does). Also, many species' coats of fur are just too short to adequately conceal the features of their genital or anal area in the first place. Anyone who has seen a male pig or non-neutered male dog, for example, will have no doubt noticed the prominent pair of features between its rear legs.
The sole exception to this censorship is the four teats of the female cow (and also quite often the male cow), which are always present, thanks to the iconic image of the country milkmaid. The only time cow udders were ever censored was in the early 1930s, when Moral Guardians became offended by the showing of Clarabelle Cow's bare udders in the 1930 Disney short "The Shindig." Other female mammals, even goats, are almost never shown with explicit mammaries, even in scenes where her offspring are shown nursing from her.
For the purposes of defining examples, only animals who are mostly or "completely" normal in physical appearance (e.g. not visibly anthropomorphized) and animal mannerisms should be considered. For cases where a creature is sufficiently anthropomorphized that the Uncanny Valley effect might occur (such as with Funny Animals in a few cases and especially with Petting Zoo People), see Nonhumans Lack Attributes instead.
Sub-Trope of Nonhumans Lack Attributes, and related to Barbie Doll Anatomy (the human version). Compare Funny Animal Anatomy. See also Appropriate Animal Attire for a broader discussion of modesty and clothing as it applies to animals.
Anime and Manga
In the Ginga Densetsu Weed anime, every dog is drawn this way, despite the fact one character is known for biting the testicles off of his enemies. This is sometimes averted in the manga though, as some panels clearly show genitals and anuses.
Averted in many anime appearances of Tanuki, raccoon-like creatures that are often depicted as having gigantic testicles as a symbol of luck.
In Naruto, at least with a certain giant armadillo.
Zig Zagged in What's Michael and Jarinko Chie: testicles (but nothing else) are shown.
Averted in Bone with Roque Ja the mountain lion, who when seen from behind has a very prominent scrotum.
The anus part of this trope is averted with the comic book version of The Smurfs cat, Azrael.
Averted by volumes in Pom Poko, where the testicles of the tanuki are very prominent, both art- and plotwise — though this is more due to the fact that tanuki having a very large scrotum is an aspect of Japanese folklore than any bizarre decision on part of the director.
Averted in the movie versions of The Plague Dogs The dogs as well as most other animals are drawn anatomically correct.
Zig Zagged in 'Watership Down''. Although the dogs are anatomically correct, the bucks are generally not drawn with anything showing no matter how close the camera comes. With dogs, on the other hand, it's quite obvious.
Truth in Television, male rabbits are very hard to distinguish from female rabbits, the differences in the two sexes' external genitalia are very subtle.
Extremely applicable for the rats from Ratatouille, as male rats normally have testicles as big as their snouts. When Remy walks upright, his scrotum really ought to have been dragging on the ground.
Played painfully straight in Dumbo where all of the circus animals even have to rely entirely on Delivery Storks in order for them to have offspring.
Poor Bruce from Finding Nemo was originally animated with prominent claspers, as would be appropriate for a shark his size. Obviously they didn't make it to the final version.
In the 1933, 1938, and 1976 versions of King Kong, Kong had no nipples.
Lassie is a female dog (note the root of the name is Lass not Lad) played by males with their belly hair coiffed very carefully to cover up the naughty bits.
Harry and the Hendersons played this one straight. Harry's fur is a bit thicker in that area, but such doesn't really conceal genitalia in hominids.
In Cats & Dogs, Lou is played by a couple of different beagles. Any scene in which he rolls on his back (particularly if someone is scratching his belly) will be played by a female, who naturally has less in that department. Despite this, Butch's comment on his immaturity is "He's still got his you-know-whats, for cryin' out loud!"
A particularly curious example is in Son of the Mask where Otis the dog is partly played by a real dog and partly by horrifying animation. When the dog is real, it obviously has the equipment you would expect but as soon as it puts on the mask and becomes animated the downstairs department is noticeably lacking. This could have been Hand Waved as being an effect of the mask itself, had it not been for the fact that the whole premise of the movie is that a human male impregnates his wife whilst wearing the mask.
The 2010 version of Alice in Wonderland has the family of bloodhounds. The mother of the puppies has no discernible mammaries, and the father has nothing between his legs with which to create said puppies, even when he flops down on his side in front of the White Queen with his crotch pointed right at the camera.
In Rise of the Planet of the Apes none of the apes have any genitalia. Chimpanzees have gigantic testicles, it's impossible to censor their parts without it being obvious.
Averted with the tiger, Richard Parker in Life of Pi as he is rendered with testicles.
This trope is invoked in-character in Thief of Time, when Susan takes a paper tube away from Jason and informs the boy that, no, the cardboard animal which the class is constructing is a polite horse.
Possibly a reference to a 19th century joke campaign to clothe animals: "A nude horse is a rude horse." was their slogan.
Also from Discworld, the illustrations in The World of Poo include rear views of Widdler the dog and of a rabbit. Both avert this trope by including a visible dot for an anus, which is only to be expected in a book all about what comesout of one.
Very explicitly averted in Neuropath when Tom's dog rolls over and exposes his genitals. Agent Gerard find it hilarious and says it's so big it should get its own website. Then makes up some domain names. ("www.dog-got-a-bone.com")
Myths & Religion
Japanese statues of the tanuki feature the animal with large testicles but a very small or completely lacking penis.
Bryer model horses, off the shelf anyway. The lack of genitals is the first thing that hobbyists customizing their models address.
Small pet rodents in films or television are virtually always played by females, even if the animal is male in-story. This is because male hamsters, rats, and mice have very large testes that would be virtually impossible to hide.
In heraldry, male mammals are normally depicted with a penis if their legs aren't in the way. If the penis is intentionally omitted, the animal is described as "éviré".
For some reason, this trope is commonly averted on Deviantart, though the attributes are not necessarily in their proper places.
Some dog lover found the Dog's lack of attributes so annoying in Dragon Age: Origins that he made a mod that put a modest sheath between his legs... and was promptly accused of interest for bestiality by the community members. This trope is strong in people's minds.
The fact that Pokémon lack visible attributes (with exception for Miltank's cow udders, of course) is likely a culmination of its simple visual art style and kid-friendly target audience (and G-Rated Sex). The actual Mons can probably be assumed to still possess their attributes...after all, that Berry fertilizer (made in part from their "well, you know...") has to come from somewhere....
Various Pokémon draw their designs from things other than animals, so whether their attributes could even be identifiable as such varies as much as the Mons themselves. How exactly does one tell the difference between, say, a male and female Grimer — living blobs of toxic goo? Eh, forget we even asked.
Can one tell between male and female "living blobs of toxic goo"?
Bidoof might be an aversion. The male has two extra bumps on the underside of his tail.
Partial aversion in Final Fantasy XII—no monster has genitalia (or anything we would recognize as it), but a few do seem to have anuses—especially the wolf monsters.
The wolves in FFXII actually do have female genitalia. Interestingly enough, the giant wolves have a different texture from the regular, and they're noticeably in heat.
Knights of the Round has the Mad Tiger enemy, which when knocked into the air is revealed to have a sizeable pair of balls.
Mostly played straight: The other animals in the realistically rendered video game, Red Dead Redemption don't have any genitals but the cattle and horses do (if you look very closely in the horses' case).
Partial aversion in Tales of Vesperia. Repede has a visible sheath, but no testes, and other dogs, as well as the monsters, are noticabley lacking in the downstairs attributes.
Zig-zagged in A Dog's Life. Jake has a noticeable sheath in the intro animation but in the game itself he's bare.
Amaterasu, the main character of Ōkami, has a spot under her tail that appears to be an anus, averting this trope. However, it gets zigzagged, as she doesn't have any sort of indicator of gender (no vagina or nipples).
Mr. Tibbs, a cat with a minor role in The Lost Crown, walks with his tail held high in a few scenes where his lack of an anus is apparent.
Animals in Zoo Tycoon 2 poop regularly, and some even squat or lift their tails to do so. Most (except for hippos) have no sign of an orifice for them to poop with, however.
In the "Extinct Animals" expansion pack, aurochs cows have visible udders, but no teats.
In Code Lyoko, Odd's dog Kiwi is most definitely an aversion, as evidenced by the two lumps between his hind legs.
The My Little Pony: Friendship Is Magic episode "The Best Night Ever" played with this trope. When Spike wants to be let into Rarity's boutique while the ponies are getting ready for the Grand Galloping Gala, she objects by stating they're "getting dressed", to which Applejack immediately points out that they normally don't wear clothes anyway.
In the episode "The Super Speedy Cider Squeezy 6000", Rainbow Dash wakes up Fluttershy early so that they aren't too late to get cider. Rainbow Dash pulls down the covers she's holding up over her, causing her to blush, even though she has no visible genitals (and they're all almost always nude anyway).
While generally played straight in Samurai Jack, one scene in the final episode required Jack to find milk for a lost baby boy, leading to a few depictions of female animals with visible udders, although his attempts to milk the wild animals were not entirely successful.
Zig Zagged with the miniature horses in one episode of The Oblongs. The horses lack genitals, but they have anuses that look like the letter "x.
Averted with the dog Scottie, who has both a visible anus and sheath. No testicles are shown, but since he's a dog, he could just be neutered.