It can be a habitational name, meaning someone who lived by a "finch" (a type of bird) or near a place called "Finke," or it could be an occupational name for a "finch catcher" or "financier."
It's important to note that while German origin is the most common, the name Fink can also be found in other countries like the United States, Canada, and the UK, due to migration and historical connections.