Cats do not usually have that annoying “red-eye” effect…
…but they do have green-eye (or white-eye, blue-eye: anything but red.) Why?
Because they have a layer of reflective cells, right behind the retina. The “tapetum lucidum” (bright carpet) helps them make the most of whatever little amount of light there may be, as they hunt for mice in the pitch dark desert. It is because of the tapetum lucidum that cats can see more than humans in the dark.
Here’s a close up:
So that’s why cats have “green-eye” instead of red-eye, when you shine a light (like a flash) into their eyes. And it is especially in the dark, when their eyes are open to the fullest extent (when, as you see, they have huge pupils), that this shows.
Note that Lightroom’s “red eye removal” tool does not work on green-eye. Best to avoid it in the first place, then; and the way to avoid it is to avoid shooting with the flash near the lens, especially in the dark. No pop-up flashes for kitty!