For instance:
Person with kid: "Oh, my kid makes the biggest messes, it's so annoying."
Person with no kid but pet: "I totally get that, my dog will just tear up my apartment if I'm not paying attention."
Then the person with a kid gets bothered because the person without a kid but with a pet is equating their pet to a child. I think people just want to try and relate with others. When they don't have a kid, but they have a pet, they try and use that to make connections with people who have kids.