Not cats as shields but animals like cats and dogs were painted on the shields in the likeliness of the Egyptian’s goddess. Still worked and made them run away. They then later threw cats in the faces of the captured Egyptians, mocking them for losing a city to protect common animals.
The reason we have cats is because our ancestors saw cat ancestors eating their garbage and thought "awww they're so cute! And too small to kill me!" And the cat fandom grew to a point a group of people worshipped them and this all happened before the internet.
Are we certain that the cats weren't just operating the Persians like Mecha warriors?