Hi!
The association is enforcing the Riparian Water Rights.
The Riparian Water Rights allows the landowners to make reasonable use of a body of water which flows through their properties. These reasonable uses include fishing. If the water is non-navigable, the owners have the right of exclusive use of the body of water.
When an association in a town shares a lake, and only the association's members are allowed to use the lake for fishing, the association is enforcing the Riparian Water Rights.