Nice double-alligator.
Most "alligator" wrenches, which came in a multitude of sizes, from very small to very large, had one side of the jaw serrated and the other smooth, and were intended for pipe, secondarily for fasteners. If you look at old catalogs, the tables would include a column for the pipe sizes each would handle. 4.c's ads illustrate the same point.
Like Crescent, Kleenex, and Xerox, the name was colloquialized from a brand name. You won't find the word "Alligator" on any alligator wrench but one, those made by American Saw Works, which trademarked the term in 1887, or Roebling, which bought out ASW in 1901. Bonney literally called their alligator wrenches "Crocodile" wrenches, among other names, to avoid infringement. EDIT: More on that
here.