Maybe the fish should be the property of society and the location of the fish is the catchers right, with safe handling and responsible returns being their responsibility?
I don't know what you mean by responsibility here. In the current system fisherman are incentivized to catch fish by the profit* from selling them, which they pay taxes on. You need incentives because picking people and forcing them to be responsible for fishing would be slavery
"profit" is a different incentive than divine revelation although both have worked in human history.