The fish didn't chew the nails off; they just damaged the nail plate, probably near its root. I found the story at Gizmodo:
According to a case report published by her doctor in JAMA Dermatology, the woman’s toenails stopped growing and started falling off soon after she received a so-called fish pedicure... Six months into her nail troubles, she visited a dermatologist, who ruled out any known causes of onychomadesis, such as major illness or a side effect of certain medications...Photo via The Soul is Bone
“I do not recommend fish pedicures for any medical or aesthetic purpose,” Lipner said. “In addition to onychomadesis, there are also serious infections associated with fish pedicures.”
addendum for the wordsmiths: That's a new word for me. Etymology onycho (nail) + madáō, “I fall off”(Greek). Is there any other English usage of the madáō/madesis term???