Fishing quotas that are enforced by regulatory bodies are what still prevent it from running rampant within the waters of said organizations.
The free market does not just mean "anarchy! take everything!" There's no market here!
If you're resorting to arguing the semantic meaning of Free Market, then it is definitively "an economic system in which prices are determined by unrestricted competition between privately owned businesses."
With natural resources, the free market is certainly an unrestricted first-come first-serve market. Fishing quotas are restricted competition and price influential, which makes it not a free market, despite your attempts to redefine it as such to fit a narrative.
A market solution would allow a fisherman to catch below quota and be rewarded in later years by replenished stocks in the waters he controlled.
This is a problem with some of those quotas - they're not being set up right. That's the failure of democracy though, not of central planning per se.
> A market solution would allow a fisherman to catch below quota and be rewarded in later years by replenished stocks in the waters he controlled.
That may work, in a type of business with big inertia (i.e. where you can't go out of business over a single season), if we could parcel water like that. Sadly, fish colonies don't respect arbitrary lines we draw on maps. I don't know if such a market solution is ecologically possible; fish need space, and they often need to travel.
What if I buy a 1 meter by 1 meter parcel of the ocean and charge bottom of the barrel prices to dump industrial waste in it? What if that square of ocean was 100 meters off shore from a beach in Los Angeles? If you don't want that then we're right back to government regulations.