Are “Rings” in fish like rings in a tree?
It depends which structures you're talking about. However, several body parts of bony fish such as the scales and otoliths (earbones) generally contain a record of seasonal growth patterns. So in that sense yes, the process of ageing fish is similar to finding the age of a tree by counting rings in the wood. For different species of fish, different structures are used to determine ages.

https://www.nefsc.noaa.gov/fbp/basics.htm