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I'm not a big zombie afficiando so excuse my ignorance here.

In those stories where zombies need human brains for sustenance are they only able to live by cannibalism, or would the brains of other mammals or animals suffice?

Sufficient but undesirable -- like eating a McDonald's hamburger when you're craving a thick juicy steak.

but the long answer is that I imagine it would depend on the writer. I know that in Anne Rice's universe, vampires could basically suck on animals as if they were boxes of juicy juice to survive, but in other universes of the vampire mythos it's not possible at all

I haven't encountered any stories where zombies needed anything for sustenance, so I wouldn't know. Most stories tell of them craving flesh or brains, but not requiring them. Usually however, it's only humans.

In World War Z/Zombie Survival Guide zombies will eat any living flesh. There's actually an anecdote about the Bravest Astronauts in the World (TM) watching a zombie on Earth spending days trying to dig up a gopher or something.

In these books sustenance isn't required, digestion isn't happening, and zombies will just keep eating until their stomachs burst.