The claim is easy to test and has been tested repeatedly. Goldfish trained to associate a sound or light with food continue to respond after long gaps without training. In one widely replicated school-level protocol, fish learned to feed only when a lever was pressed at a particular hour and retained the timing across weeks. Other work has demonstrated maze learning and avoidance conditioning with retention measured in months.
The mechanism is not exotic. Teleost fish possess a telencephalon whose lateral pallium is functionally comparable to the mammalian hippocampus in spatial learning tasks, and lesioning it impairs place memory in the ways that prediction implies.
The larger error in the myth is what it licenses. A fish that can learn, remember and anticipate is a fish whose housing conditions matter, and goldfish kept in unfiltered bowls are commonly stunted and short-lived when the same animal in a pond may live well over a decade.