In the US people that own pets seem to be pegged as either dog people, or cat people. This cultural phenomenon is so prevalent that Hollywood even made a movie about it.
My question is, when did this rivalry begin in the US?
According to Katherine Grier author of Pets in America: A History (2006, UNC Press, arguably the authoritative work on the subject), the sentiment is probably rather recent, since the 1970s.
I will first make a couple of quick points:
The notion of preferring a dog to a cat or vice versa is only possible in a society where both are more familiar as companion animals than as working animals. After all, for most of human history, most domestic animals served an economic purpose: cats were for controlling household pests, dogs were for hunting or herding duties and guarding on the side. Songbirds filled the air with music before there were records or radio. Preferring the company of one would not have ruled out owning the others as well.
Grier notes that pet-keeping became mainstream among the bourgeoisie only in the mid-19th century (and dogs were kept purely as pets before cats were). Most pet owners would moreover have interacted with animals in traditional ways both inside and outside the home; not only horses, but pigs and chickens could be found in American cities into the 1920s.
A surge in ownership took place after World War II as prosperity and suburbanization enabled more families to keep pets. This neatly coincides with the earliest Google Books search results for "a dog person" and "a cat person" come from the late 1950s and early 1960s (aside from references to some Hollywood sci-fi/horror species).
Grier notes further that since the 1970s, "the practice of pet keeping has evolved at an accelerated rate." Spending on pets (toys, food, veterinary care, training, "doggy day care" and so on) has seen a sharp uptick, and perhaps the emotional energy available from declining marriage and birth rates has been redirected into sentimentality toward the animal population— the term “pet” itself is uncomfortable for some. She cites a figure from the American Pet Products Manufacturers Association that even 20 percent of reptile owners describe their pets as being "like a child/family member."
Anecdotally, this seems to bear out. I don't know a single cat owner who puts Morris out for the night as depicted in the closing credits of The Flintstones. Witness also the firestorm over Mitt Romney bringing the dog along in a rooftop carrier on a family road trip in the 1970s. Like driving without a seat belt, such practice was quite normal then, but today invites clucking and appalled shudders from the chattering classes.
With so much invested— financially and emotionally— into pets, it should be no surprise that we project our identities onto them as well. While a crude measure, the Google NGram of 'a cat person', 'a dog person' shows a significant uptick in the meme after 1980.
According to Google ngrams, the terms started to take off around 1970. Early uses of "dog people" and "dog person" mostly seem to be referring to people who have expertise about dogs (e.g. a 1915 article in Dog Fancier that refers to judging shows by "dog people"), but by the early 1970s the terms are clearly being used mainly to indicate people who just prefer one or the other.
The earliest version I found that clearly uses them as an "us or them" comparison was from a 1919 article in Everybody's Magazine (The Case of Mouser vs Bowser) shows that the concept was already out there that early; the table of contents teases the article with "Are you a cat person? A dog person? If so, why?" (The actual article uses "dog-person" and "cat-person" instead of the non-hyphenated versions.)
After that, the next appearance I found was from 1955 (Arts Magazine, Volume 29, Issue 3): "Are you a cat person or a dog person?" she asked. "I think there are two kinds of people." There's also one from 1963 ("The Whistling Zone", by Herbert Kubly): "To me everyone is a dog person or a cat person," she was saying, her hand on his arm.