Here is a list of cat songs by year, and a list of dog songs too. I only counted references to domestic animals (dog, puppy, cat, etc.) rather than wild species. And I only counted original rather than cover songs, to show what mood was like at the time of naming the song. Grouping them by decade gives the table below.
Decade - Cat ___ Dog songs
50s - 1 ___ 4
60s - 12 ___ 12
70s - 8 ___ 7
80s - 3 ___ 1
90s - 2 ___ 1
21c - 1 ___ 2
Dog songs outnumber cat songs in the '50s, then suddenly during the '60s cats rise to equal prominence. They just barely overtake dogs in the '70s, though cat people really come alive in -- when else? -- the '80s. They appear to outnumber dogs in the '90s, though cats only appear in the first half of that decade, and the dog song in the second half. Dogs are clearly back in the lead during the 21st century.
To help clear up the picture after the '60s, when each decade doesn't have a whole lot of animal-named songs from which to judge definitively, we can also look at names of the recording artists themselves. There was Cat Stevens who had a string of hits in the '70s (and "Cat" is not a short form of his given name, or middle name). Then there were the Stray Cats who had several hits in the early '80s. MC Skat Kat sang a duet with Paula Abdul in the late '80s. Heavy metal bands from the '80s chose wild cat names -- Def Leppard, Pantera, White Lion, White Tiger -- but not wild dog names. (They chose other animals, just not dogs -- Ratt, Whitesnake, Jackyl, etc.)
Dog names show up more during the '90s and 21st century, primarily with rappers. Snoop (Doggy) Dogg, Nate Dogg, Pitbull, Lil' Bow Wow, and so on. The only exception is the Pussycat Dolls from the mid-2000s (part of the '80s revival at the time?).
Side note: black people are not cat people.
The rise of cats in pop music tracks the violent crime rate and the outgoing phase of the cocooning cycle. Why? Beats me. Cats are more aggressive and assertive than dogs, and respond more to tenderness than to fear, so folks from the '60s through the early '90s would've identified with them more as a totem animal. People were also in a more conservative mindset, as far as wanting to preserve what was good in the past, and having a heightened sense of disgust, which would tilt them toward cat people.
Why do cocooning times bring out the popularity of dogs? Pop culture gets kiddie when the crime rate falls, and dogs are more cutesy than cats. ("How Much Is That Doggie In the Window?") Dogs are good for wannabe thugs who front like they're badass, a more common thing in falling-crime times when they won't be put to a real test. Dogs also allow socially awkward people to interact in a mediated and thus more comfortable way -- through their dogs, e.g. at a dog park.
You know, it's not necessarily that they're socially awkward. It's more like people need some kind of cover story, rationalization, and plausible deniability in a climate of suspicion and cocooning. Being openly outgoing is creepy and suspect, but if you're just taking your dog out for a walk, and happen to meet up in a place where there are other people taking their dogs out for a walk, well, it's not like you were trying to go out and socialize, so you're probably not a creepy crypto date rapist.
Does this mean that folks who came of age during the '70s, '80s, and early '90s are more likely to be cat people? I could see Generation X being more cat-loving than the Millennials, who strike me more as dog people, as do the Silents and early Boomers, while late Boomers could go either way (they seem slightly more cat people-y to me, though).
I'll be running through the history of the popularity of cats vs. dogs in other areas of pop culture later on. I wouldn't have thought that their relative popularity could cycle, but why not, if just about everything else does?