Sapient All Along

In real life, there are mental differences between types of animals that go beyond incompatible vocal abilities. If we managed to make an animal translator in real life, they would not be able to communicate and interact exactly like humans. Not so in fiction, in an otherwise realistic setting, the ability to talk to animals will often result in every species having the same mental capabilities as humans, which they apparently always did. You'd think they'd try harder to communicate with humans if this is the case, but they're content to wait until a human bridges the language gap for them.

Can lead to Fridge Horror, if all animals were sapient the whole time that means pretty much every human civilization in history has been constantly doing things that amount to slavery and genocide. The animals don't seem to mind though, their first reaction to talking to a human will never be begging them to spread the knowledge that animals are mentally equal to humans. They may even want their knowledge to be kept secret, despite this having no noticeable benefit to their species.

Comic Books

 * The Legion of Super-Heroes had this in the Silver Age with the Legion of Super-Pets. Proty, a shapeshifting blob, was capable of not only turning into Superboy's shape, but speaking to the point where people could think he was Superboy.  He could also communicate with all the other Super-Pets telepathically on a level, implying that they were all similarly sentient.  Including Superboy's dog.

Fan Works
"Harry stared at Draco. "You mean just magical snakes, right?" "N-no," said Draco. He was looking rather pale, and was still stammering, but had at least stopped the incoherent noises he'd been making earlier. "You're a Parselmouth, you can speak Parseltongue, it's the language of all snakes everywhere. You can understand any snake when it talks, and they can understand when you talk to them... Harry, you can't possibly believe you were Sorted into Ravenclaw! You're the Heir of Slytherin!" ... ... ... "SNAKES ARE SENTIENT?""
 * The fanfic Harry Potter and the Methods of Rationality gives us this exchange between Harry and Draco, after Draco learns by accident of Harry's Parselmouth ability (which, as in canon, Harry himself was unaware of).

Films - Animated
"Woody: (to Sid)We toys can see everything... (he spins his head around 360 degrees) ...so PLAY NICE!!!"
 * The Princess and the Frog demonstrates this trope when the two main characters are turned into frogs. Apparently there was even an alligator that learned to play the trumpet, but never thought of using his remarkable hand coordination to try writing down that he was sapient.
 * Pound Puppies: The Legend Of Big Paw takes place in a world where dogs and humans gained the ability to communicate in the middle ages, and it seems to be common knowledge among humans. This gets a little disturbing when you realize that despite full knowledge that dogs are as intelligent as humans, they are still treated as pets.
 * Averted in Cloudy with a Chance of Meatballs. Flint develops an animal translator and places it on a monkey named Steve (voiced by Neil Patrick Harris). Steve is still depicted to have the intelligence of a monkey, being incapable of forming complete sentences and not being able to say more than his own name and whatever object is in front of him at the time.
 * The toys from Toy Story.

Film - Live Action

 * Doctor Dolittle, more The Remake than the original, where we actually hear the animals talk.
 * Variation with Tron where instead of animals, it's computer programs. If you get digitized into a computer you can have a normal conversation with a word processor.

Literature

 * Averted in Children of the Red King. One of the characters can talk to animals, who generally have less of an understanding of the world than the humans.
 * In one of the Wizard of Oz books, the characters wonder why Toto is the only animal in Oz who never talks. He reveals he could talk, but was afraid if he did, Dorothy would stop treating him like a pet. And he kinda liked the pampering.
 * In the Known Space series, dolphins turn out to have been sapient all along, and are granted the same rights as humans in the 1990s. Humans develop a thriving industry in selling artificial hands to dolphins, and they sue us for whaling.

Tabletop Games

 * Usually averted in Dungeons & Dragons. Most spells or other magic used to communicate with animals state that the animals' speech should be limited by their lesser intelligence, so you aren't going to be able to have a philosophical discussion with a dog.

Western Animation

 * The Wild Thornberrys may be the most thorough example in terms of demonstrating that every single animal is actually sapient, each episode has a new location and group of species demonstrating this to Eliza. Environmental protection is always promoted, but not recognizing animals for the sapient beings they are.
 * The PBS show Arthur eventually revealed that babies and animals are able to communicate, and both secretly sapient. Even insects were eventually shown to be sapient.
 * Subverted in Dexter's Laboratory, where Dexter creates a device which allows a dog to talk, only for the dog to just repeat things like, "Look, there's the thing! It's a thing! Look! A thing! There it is!" Further demolished when the dog's owner shows up, and is just about as smart as the dog.