Agree with James. The two species of bacteria we need in our filters are the same for freshwater cold fish (such as goldfish) as for tropicals, so the mature filter media can be transferred.
Using the term "washing" can be dangerous with beginners. As said, filter media is rinsed in tank water. The process is to gently squeeze sponges or gently swish trays of other media such that you are getting much (but not all) of the loose debris out of the media to unclog it. You definately don't want to strongly wring out sponges or take other aggressive cleaning actions, because you want to preserve as much of your bacteria and their biofilms as possible. By "mature" we mean filter media that's been running for 6 months to a year (it takes about a year for full maturity of a filter.)
When combining some mature media with some new media it always works better to put the mature media earlier in the water flow path or to intersperse it. The act of using a significant amount of mature media to start a new tank is called "a clone." Ideally the filter is then "qualified" via a short household ammonia fishless cycling process to confirm that the transfer was a "take" and that everything is working properly before exposing fish to the environment.
~~waterdrop~~