And not property rights the way water rights sometimes works, where a group of people all have the right to take all they can.
If the same people owned an aquifer but all had to pay for what they used, the water would be more properly priced.
With water, as with fish, people often pay the cost of taking the resource and not a price that reflects the value of the resource.
The value is actually more like the future cost of replacing the resource. Once your aquifer is empty, you must ship water in. Or desalinate. That's more like the real cost.
With a species of fish, the replacement is incalculable. But one thing is clear. The price should increase rapidly as it becomes scarcer, preserving the incentive to keep the population viable.
That can't happen if people just pay the cost of going out there and getting the fish. The price has to reflect the value of the fish too.
If the right too harvest a population were a piece of property that could be bought and sold, the incentive to preserve the population would be stronger.
You might even see technologies that avoid taken pregnant females and that count how many males there are so you have enough of those too.