Where did "doggy dog world" come from?

"Doggy Dogg World" is a Snoop Dogg rap from the early 90s, about the time the phrase picks up in the Google Ngram. The phrase appears to have been adopted by the dog rescue community and, yes, as currently used there it appears to be an eggcorn.

In a previous version of this answer the name of the rapper Snoop Dogg, born Calvin Cordozar Broadus, Jr., was spelled Snoop Dod. I regret the error.

The Ngram chart in the OP's question suggests that "dog eat dog world" first appeared in print in 1954, and that "doggy dog world" first appeared in 1984. A Google Books search, however, finds earlier instances of both phrases.

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1. In the jungle out there, do dogs normally eat dogs?

In fact, the phrase appears to have arisen first in the context of an observed disinclination of dogs to eat dogs, as Christine Ammer, American Heritage Dictionary of Idioms (1994) points out:

dog eat dog Ruthless acquisition or competition, as in With shrinking markets, it's dog eat dog for every company in this field. This contradicts a Latin proverb which maintains that dog does not eat dog, first recorded in England in 1543. Nevertheless, by 1732 it was put as "Dogs are hard drove when they eat dogs" (Thomas Fuller, Gnomologia).

But the sense of the Fuller quotation and the Roman saying are much to the same effect: that dog eating dog is highly unnatural and therefore a sign of an environment in extremis. The notion that, in human relations, "it's a dog-eat-dog world out there"—which presents (figurative) canine cannibalism as an essential aspect of the normal state of the world—is thus fundamentally at odds with the original sense of Fuller's saying.

In any case, the phrase "dog-eat-dog world" shows up in Google Books search results as early as 1918. From Elias Tobenkin, The House of Conrad (1918):

"And now, what has happened? Men have come here and have transplanted, and adapted to new conditions, the Mephistophelian system of exploitation which is current on the old continent. They have built factories and shops here, they are operating mines and mills, no to keep the wants of all supplied, but for personal enrichment. They operate not on the principle of justice but on the principle of might. They turned this country into a dog-eat-dog world. The wages they pay are arbitrary and the worker must live accordingly. If he fights back he gets a little more; if he is quiet they may try to reduce his meager stipend. In a land which is the youngest country on earth we have more windowless rooms, greater congestion, more poverty, consumption, crime, insanity than are to be seen in the cities of the Old World. ..."

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2. When and why did 'doggy dog world' emerge?

As for "doggy dog world," the first Google Books match is from U.S. Senate Committee on Veteran Affairs, Educational Benefits Available for Returning Vietnam Era Veterans (1972) [snippet]:

"What made Nam so beautiful, in a sense, was that people there worked together. You could depend on your buddy. But here it's a doggy-dog world."

There was sadness, perhaps even a trace of bitterness, in the voice of Jon Buchanan, a veteran of 15 months in Vietnam and a drop-out of the Police and Fire Department's training program for minorities. He is now unemployed.

The rise of "doggy-dog world" in the mid-1980s seems to have occurred independently of the instance recorded in 1972. Andrew Tobias, Money Angles (1984) [combined snippets] introduces the wording in the context of a child's misunderstanding of the phrase "dog-eat-dog world":

[Chapter] 7 KEEPING IT

A Few Words on Taxes from a Man Who Knows

Somehow, in this doggy-dog world, you've made it. ("It's a dog eat dog world out there," I warned a six-year-old. "It's a doggy-dog world out there!" he giggled back gleefully for the rest of the evening, entirely missing my point.)

To similar effect, from an item on phrase misunderstandings in New York magazine, volume 17 (1984) [combined snippets], we have this:

Among the perfectly fetching entries we've seen before in this category: A blessing in the skies. Shirley, goodness and mercy. Gladly the cross-eyed bear. A washed pot never boils. It takes two to tangle. Money in escarole. Transcendental medication. Youth in Asia. It's a doggy dog world. The Winchell factor. Stand beside her and guide her through the night with a light from a bulb. A new leash on life. It will cost you a nominal leg. Poultry in motion. Your name in headlights. For all intensive purposes. The fat is in the fryer.

And from The Esquire Fiction Reader (1985) [combined snippets]:

She said Louie was thinking of selling them [Italian carpets] in Idaho. She said — and this would've made you laugh, Eddie, it would've truly — she said it's a doggy-dog world out there. Doggy-dog. She was real cute. When she said that, Louie got down on the floor and barked like a dog.

"It's a doggy-dog world" is thus simply a rewording of "It's a dog-eat-dog world" in misheard and misparsed form. The 1984 upsurge in published occurrences of the phrase probably reflects not a sudden spike in real-world confusion over the wording of the phrase, but either coincidental observations of the mistake or the spread of such observations from one published source to another. The matches from Google Books do not include any appearance of "doggy dog world" in a bestseller or other source that is likely to have triggered mass awareness and usage of the phrase.

It also bears observing that in 1984, the year when, according to Ngram, a measurable frequency of use of "doggy dog world" first arose, the phrase "dog eat dog world" was roughly ten times more common, whereas in 2000 (the last year tracked in the Ngram chart), "dog eat dog world" is now almost twenty times more common. So in relative terms, the success of "doggy dog world" over the first sixteen years following its abrupt arrival on the scene was less than overwhelming.