It's actually very simple....and it depends on the fish.
If the fish is buoyant, its density is less than the water. This means that its body weighs less than the same volume of water.
If the fish is negatively buoyant (i.e. sinks), it's more dense than water for the given volume. It has more mass than the given volume of water.
So a fish with a bloated swim bladder weighs less than the water it displaces; and something lacking a swim bladder (or with a vestigial one) weighs more. A pleco would be an example of something heavier than water.
Salt water weighs more than freshwater (it's more dense), so if you throw a freshwater fish in a saltwater tank it will often float.
Another example: Most people with average BMI will float in fresh water when they hold their breath (an increase in volume that marginally increases weight), but will sink when they exhale (less volume)
And people are forgetting that density = mass / volume
So when these people are saying that density is independent of weight, they are horribly misguided.