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.
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").