Yes, this is correct. An established tank that has a fully cycled filter and has been running for at least a number of months will have a bacterial biofilter that has matched the bioload (the fish and other creatures) and the two (the creatures and the filter) are portable, they can move -together- to any new water volume (such as a new tank.) The old filter will continue to handle the old fish (no new fish or creatures should be part of this process of course.)
There will be a slight loss of overall numbers of bacteria because some of them live on the old tank walls and gravel but these are quite insignificant usually, compared to what is in the filter, so the moved filter will still handle the fish and will quickly move slightly upward in bacterial colony sizes.
Meanwhile, the new filter will (extremely slowly) begin to get a few bacteria become a part of the overall biological filtration. Eventually, after about a year, the two filters will have balanced 50/50 or so if you were to leave them both running. The balancing of the two filters can occur faster if you transfer maybe 25% of the old biomedia over to the new filter, still running both for a long time. Never remove more than 1/3 of the biomedia from an established filter that still needs to care for fish.
Now there are other, completely different ways to accomplish the overall goal of course. Instead of sharing filters on the new tank, you can simply seed the new filter with 1/3 of the biomedia from the established filter and then perform a fishless cycle to qualify the new filter. The old filter will quickly enlarge again and still be taking care of the established fish in the established tank. Meanwhile you will be fishless cycling the new filter with ammonia and playing around with the decor of the new tank without the fish being in there yet. This is also quite a popular procedure among experienced aquarists.
~~waterdrop~~