Aside from the repetitive appearance of the black cat, I can't remember seeing any living animals, either inside the Matrix simulation or in Zion.
How come? Is there an in-universe explanation?
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Although we do see animals inside the Matrix on numerous occasions (see below), the word of god answer is that all of these animals are simulated:
TheTrinityACMXCL: What about the animals in the Matrix? Are they real animals or just computer-generated images?
Wachowski Bros: Sadly, they are all computer generated images.
This was backed up by Phil Osterhoose (Assistant to the Wachowskis) in The Art of the Matrix
A lot of people have asked if the animals in the Matrix are also hooked up in pods. Unfortunately, they are not. They're all computer-generated images.
In addition to the "Woah, deja-vu!" cat seen in the first Matrix film we also see a large number of crows in The Matrix Reloaded.
As well as a dozen or so live fish.
and a flock of birds in The Matrix Resurrections
Kid's Story features some more crows and an (unnamed) cat.
There's a dog in World Record.
There are also doves seen in the Animatrix story Beyond as well as another dog, some butterflies, a considerable number of rats and two cats; Yuki and another unnamed.
A Detective Story contains another cat (named Dinah).
The Animatrix story Matriculated contains what appears to be a cybernetically enhanced monkey named Baby. This is especially noteworthy because the story takes place outside the Matrix, providing evidence of the survival of higher mammals.
In the Matrix comic Broadcast Depth, one of the characters encounters an octopus eating a crab as well as a large school of fish.
The central character of the Matrix comic Butterfly is arguably the titular butterfly.
Deja Vu mentions an (unseen by us) infestation of ladybugs and we also see an American robin.
Let It All Fall Down features a woodland scene sporting an eagle.
Run, Saga, Run features a white rat.
Sweating the Small Stuff includes a cat and a spider.
The Miller's Tale is again notable because we see wildlife surviving outside of the Matrix. This includes small mammals, ducks, frogs, insects, arachnids and slugs.
and from A Sword of a Different Color, we see someone eating what appears to be uncooked rat.
The original concept art by Geoff Darrow contains a hungry mouse.
And of course, who could forget that Cypher is eating what appears to be a large chunk of dead cow?