That is, an individual animal with a name, like Alexander's horse Bucephalus.
Ta-Miut, circa 1380 BCE, beloved cat of Prince Thutmose. *
Her name translates roughly to "Female Cat," which doesn't actually sound like a name, but bear with me for a moment.
Old Thutmose loved Ta-Miut so much that her body was placed in a little sarcophagus decorated with a picture of the cat sitting in front of a table heaped with offerings with the text "Osiris Ta-Miut, true voice before the Great God."
"Osiris Ta-Miut" is hard to translate exactly, but it generally means "the immortal Ta-Miut," which is the Ancient Egyptian way of saying that "Ta-Miut has gone to heaven."
Back to her boring name. Given that Thutmose liked his cat so much, why did he call her "Female Cat?"
So bland.
But maybe that's just our sterile, modern translation. Though we'll never know for sure, I've heard convincing arguments that a better translation for the name should be "Miss Kitty" or the like, which is just awesome. It's what my college roommate called his cat. Well, actually, he called her "Miss Pussums," but given the loose nature of Ancient Egyptian translations, it's possible Thutmose did too. I like to think he did.
*: I have no idea if this is the first. It's just the first I'm aware of.