Any organism with a mouth can bite. Whether they inflict a wound or injury is another question.
When someone thinks of an injury from a fish, most people would think of sharks and the reported incidents of them biting off a limb or leaving a nasty gash in one’s side. While most people believe these are just ruthless animals, the reality is that sharks, as well as most fish, are simply instinctive. The major percentile of the fish species has poor eyesight and rely on scent or magnetic pulses in the water. This being said, a surfer laying on a board can be easily mistaken for a seal, or fingers and toes can be easily mistaken for smaller fish or coral. Then the issue of water clarity comes up; is it dark? Is the water murky? Is it foggy?
However, when eyesight is not an issue, you must remember that a fish’s main goal is to survive to reproduce. So, they will do all they can to protect their offspring. A piranha is one of the many known toothed fish and is coined as a man-eater. While they do have sharp teeth, they only use them to either eat decaying matter or to protect themselves and their offspring. Most cases of piranha attacks are because a passerby wandered too closely to its nest. Other fish all over the world, even goldfish, will nip at you if they think you pose a threat to their eggs.
There are little or no cases of fish attacking a human for recreation.