An exceptionally common phenomenon in Japan, where animals are often generally referred to by onomatopoeia of the sounds they make, along with a cutesy honorific: e.g., a puppy might be referred to simply as "Wan-chan", or a cat "Nyanko". As this is slightly different from simply calling a dog "dog" (or in this case "inu"), these "names" very frequently stick and often go unquestioned.
Common name for a male kangaroo, when named as a joey: Joey.
Common name for a tomcat: Tom.
Common name for a molly: Molly.
Common name for a tabby cat: Tabby.
Common name for an orange tabby (aka ginger) cat: Ginger.
Common name for a calico (tortoiseshell-and-white) cat: Callie.
Beverly Nichols mentioned in one of his books that his Hungarian neighbours had a cat called Cica — Hungarian for "cat".
Common name for a rescued stray dog: Dog. Often, because the person who decides to take them in has been calling them "Dog" for a long time before doing so, or maybe that person is unimaginative.
Common name for a billy goat: Billy.
It's not uncommon for children to have imaginary friends or plush animals named after what they are.
There are lots of countries in the world formed by the union of states, but only one that's commonly referred to as The United States.
Subverted; unlike the Moon example above, The Sun is now a proper name for a specific star, and the generic term for a system of planets orbiting a star is now "star system." While other stars can still be referred to poetically as "distant suns," they're usually called by their proper names (e.g. Alpha Centauri, Sirius, HD 206250).
Some science-fiction works try to avert this by christening the moon "Luna" and the sun "Sol". Pretty much the same thing, really.
Adam is the Hebrew word for "man". Similarly, Andrew is derived from the ancient Greek word for "man."
By contrast, the name "Guy" has a completely different origin, but is regardless identical to the word "guy."
Colleen is the generic word for "girl" in Ireland, as is Sheila in Australia.
Donna means "lady."
George Foreman owned a dog named Doggo.
Averted, at least for some species! Research started to discover that some species other than humans actually do have names in their own languages (such as it may be). For instance whales and parrots will have specific songs they use to identify themselves with and by which others will use. Some may even end up with song 'nicknames'.
Many monotheists express loyalty to a god they call "God". Jehovah's Witnesses are notable for being Christians that avert this.
There is a shopping mall in St. Matthews, Kentucky, originally named The Mall. To distinguish it from other shopping malls in the area, it was sometimes specified as The Mall in St. Matthews. Its official name is now Mall St. Matthews.
There was also a shopping mall called The Mall in Kuala Lumpur, Malaysia. It has since changed it's name to Putra Place when a new management took over though.
Not so common, but still heard from time to time: a dog named Diojee or some variant thereof (said aloud, it becomes "D-O-G").
A survey back in the Nineties determined that the most common cat name in Britain was "Sooty," the most common in Canada was "Pasha," and the most common in the U.S. was "Kitty." This little factoid was used in Ad Bumpers during Canadian Saturday Morning Cartoons around the year 2000.
The scientific name for a western lowlands gorilla? Gorilla gorilla gorilla. Only slightly worse than Magilla Gorilla.