My mother always had a thing about cats and babies (newborns young enough to be in a bassinet). She said the cat would snuggle up to the baby and suffocate it. She is not the kind of person who freaks out about little things, so I can only think this really happened somewhere in the world one time and she heard about it and it stuck with her. We had cats when my kids were little, with no problems.
But I have a similar phobia of my own, based on a real-life experience. Our next door neighbors when I was a kid, had 5 daughters, and apparently, in an attempt to get a son, had one more. They finally had their little boy but when he was 7 months old and crawling, he found an open safety pin, swallowed it, and died. It must have perforated somewhere and they may not have even known what it was until they did an x-ray, either before or after his death (I was very young so don't know the details). Anyway, this made a huge impression on me at only 4-5 and that was my one phobia when my own kids were small. So in the end we are all shaped by our own experiences.
Many older woman have told me "the cat will suck the breathe out of a baby" whatever that means. I had three indoor cats when my babies were born. They avoided my children like crazy. Never went near them until the children were old enough to carry them around.
My friend told me a similar story. Her aunt lost a toddler by swallowing a penny that he found, they did not know why the baby died until after his death. One thing I found as a choking hazard that I did not think about was when one of my daughters started crawling and went straight to my dog's dish and started putting all the kibble in her mouth. Scared the heck out of me!