Looking at the history of the Texas agencies responsible, it looks like it started in the latter half of the nineteenth century, with some early moves done pre-Civil War, but the majority done from 1874 onward, when the Legislature started regulating fishing, and it extended to hunting in 1907.
Since you say “only for the monarch,” I assume you’re not looking for US laws alone. In Britain, the Game Act of 1831 protected game birds by requiring hunting licenses and establishing game wardens.