The saying, "Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime" is not from the Bible. It is an old Chinese proverb.
There are several other well-known and oft-quoted "Bible" verses that did not originate from that holy book. Some examples are, “This too shall pass,” “God helps those who help themselves,” “Spare the rod, spoil the child” and “Cleanliness is next to Godliness.”
Also, many people mistake certain Biblical details. Satan never tempted Eve with an apple in the Garden of Eden; the specific fruit was never mentioned. The Bible never named a whale as the creature that swallowed Jonah. In addition, the specific number of wise men who visited the infant Jesus was never identified as "three."