You know, I've heard this story lots of times, but I've never found a reliable source for it. John Kelly's The Great Mortality is my favorite general history of the Black Death, but the superstitious killing of cats never appears in that comprehensive history of the Black Plague. Similarly, Ole Benedictow's The Black Death, 1346-1353: The Complete History makes no mention of that superstitious killing.

That said, cats ─ just like dogs and other mammals ─ can and did carry the plague themselves. Boccaccio famously wrote of pigs being infected with plague boils, but there are accounts (PDF) of dogs, cats and even chickens being affected by the plague. Taqi al-Din Maqrizi provides one of the best descriptions of the plague in Cairo in his al-Suluk. He discusses animals at one point:

"There was a falconer who threw the flying birds of prey for hunting. Many birds and crows and others from these known types of birds were found dead in the field, and when they were taken from the fields and plucked, a buboe (plague boil) was found. And all the cats died until their presence was insignificant. And the news continued from the Jordan Valley … and they were finding wolves and rabbits and camels and wild asses and pigs and others among the wild animals dead and on them was the sign of the kubbah.

In some gardens of Cairo seven wolves were found that had died of the plague. And a man had four chickens that died and on each of them a kubbah was found at the region of their stomach. And a man had a monkey that was hit by the plague in its head and for three days he put water and food in front of it, but it would not take any, and it drank only once; after three days it was destroyed."

It's pure speculation on my part, but it's easy for me to believe the killing of infected animals, as described by Maqrizi and others, could have been combined with later connections of felines to witchcraft.

There are plenty of Medievalists around the subreddit, so I'll just say that cats in Europe at this time were known as pets and commonplace. The Book of Kells, for example, famously has cats playing with mice and rats.

One of the sadder stories about animals and the Black Plague comes from the Franciscan friar Michele da Piazza, who wrote about the arrival of the plague in Messina in October 1347, allegedly with a group of Genoese ships. It spread with such rapidity and violence that it turned the uninfected against the infected in a desperate attempt to save themselves. "The disease bred such loathing," de Piazza wrote, "that if a son fell ill ... his father flatly refused to stay with him."

Husbands left wives, parents left children, alone to die. Priests shunned last rites, notaries avoided making out wills in person for the infected. Only pets ignored the disease, eventually dying of it themselves. "Cats and ... livestock followed their master to death," he wrote.