It is possible to develop soft tissue and bone infections from contact with fish or contaminated water (fish tanks or aquariums, river water). There is a specific type of bacteria called mycobacterium that contribute to these kinds of problems.
Most of these bacteria are rare (e.g., M. kanasaii, M. leprae, M. abscessus, M. arupense). One of the more common causes of hand infections is one you have probably heard of: mycobacterium tuberculosis. But the one that is associated with fish is called mycobacterium marinum.
And there have been reported cases of sushi chefs developing finger, hand, and/or wrist infections from contact with fish contaminated with this bacterium.