Good question. It may be that the phrase developed from a proverb used in Roman times. This is from The Word Detective, a website that answers questions on etymology:
According to Christine Ammer's great book "Cool Cats and Top Dogs (and Other Beastly Expressions)", the proverb was most likely from the Roman man of letters Marcus Tarentius Varro. Back in 43 B.C., ol' Marcus noted that "Canis caninam non est" ("Dog does not eat dog"), meaning that even a (supposedly) lowly creature like the dog has limits, if not principles, and will not destroy its own kind.
Extended to human beings by implication, that's a comforting notion, but history tends to indicate that humans are not so principled as dogs. By the 16th century, folks were imagining a world in which metaphorical dogs did devour each other, and "dog eat dog," had come to mean "ruthlessly competitive." Not surprisingly, by the time of the Industrial Revolution, phrases such as "It's a dog eat dog world" had become commonplace.
Here's an explanation:
DOG EAT DOG -- "The struggle for survival in life or business turns man into an animal." This variation came later than the original "dog doesn't or won't eat dog," according to Gregory Titelman. Dog won't eat dog "means that people of the same type do not destroy one another. 'canis caninam non est' is Latin for 'a dog doesn't eat dog's flesh' and 'Cane non mangia cane' is Italian for 'dog doesn't eat dog.' In 1602 Shakespeare used the same idea in 'Troilus and Cressida.' First attested in the United States in 'Modern Chivalry' (1792) by Hugh Henry Brackenridge (1748-1816). Dog eat dog is of much later origin." "Random House Dictionary of Popular Proverbs and Sayings" by Gregory Y. Titelman (Random House, New York, 1996).