Riparian rights are not the same as fishing rights. The fishing rights may have been separately sold to someone else by the previous owner, possibly an angling club but not necessarily. I really don't see the problem with that. Allocating a value to fishing rights gives the owner of them an incentive to maintain the riverine environment, which is a good thing. In our tiny, overcrowded island, unregulated fishing and letting people do what they like quickly turns waterways into dead, open sewers. We saw this happen years ago to many rivers. Sport fishing, along with better environmental laws, has been a major factor in rivers, like the Thames, recovering remarkably over the past few decades.
I'm not questioning the value of a cultural prerogative, I was only trying to explain to our American members why you just can't plop down and throw a line in the river. In America, where there are plenty of rivers, the riparian and the water in the river belong to the public, not the landowner. There are such things as water rights for agriculture, but most of them were allotted on a first-come, first-served basis 150 years ago.