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Countless animals, including cats, shed tears. The question, as you point out, is whether or not these tears are emotional. Even for humans, the reasons behind crying are still somewhat of a mystery. Why did we evolve this ability? It could be because we sometimes cry when we are in pain, so the tears function as a visual signal that others may respond to. Unless the individual is a great actor, tears are also usually honest, meaning we cannot fake them as easily as cracking a pseudo smile or feigning surprise.
According to Margaret H. Bonham and Caroline Coile, authors of the book Why Do Cats Bury Their Poop: More Than 200 Feline Facts, Fallacies and Foibles Revealed, cats don’t tear up in response to emotions. Instead, cats may shed tears in response to eye irritations, allergies and clogged tear ducts, and for other eye-maintenance reasons. There is no evidence demonstrating that cats cry emotional tears of joy, sadness, pain, grief and more.
However, there are many anecdotal reports of cats crying tears in response to some traumatic happening. The emotional lives of cats are rich -- certainly more so than was suspected decades ago -- but I’m not sure cats would benefit from releasing feeling-based tears in the way that we do.
Cats reach full skeletal development when they are this old: