They have souls, if they didn't they wouldn't love us as unconditionally as they do, or have their own unique personalities.
I know my cats have souls...without a doubt.
I don't even believe people have souls. sorry, I just don't.
Do cats and dogs have personalities? Yes, they do.
I attended many spiritualist meetings over several decades, and I came to believe there is a spirit world, because I had so many strongly evidential messages from dead relatives.