- a dog barks "woof" (or some other variation)
- a frog croaks "ribbit"
is there a verb for when cats "meow" other than "meow" itself?
I am specifically looking for a word that is usually only associated with a cat's "meow" similar to how bark is usually only associated with a dog's "woof", so cry... won't work.
First off, dogs don't bark "woof". They bark "bark". They're two different sounds. Ditto yelps, arfs, bows-wow, grunts, howls, whimpers, and cries. Ditto croaks and ribbits, although note that "ribbit" is apparently American frog slang rather than a formal register.
Second, it's just a second onomatopoeia that you're using in a verb sense to describe a second different onomatopoeia in a noun sense. Bark isn't a verb employed with the noun woof. They're both nouns (1, 2) and both can be used as a verb (1, 2) meaning "to make the sound ~". Same thing with "bow-wow" (1, 2). Same thing with "yelp" (1, 2). Same thing with cat noises.
Similarly, you could as easily have a cat "mew(l) 'meow' " or "purr 'mew(l)' " or "meow 'purr' ". They could "hiss" any of them. But all of those words should more properly describe different sounds.
That said, third, there are the verbs "waul" and "caterwaul" which means to wail (i.e., meow) like a cat in heat. The OED also particularly associates "yawl" and "yowl" with cats, albeit in the same limited sense: the pained meowing of a female cat in estrus.