Not sure how long you've been a fan, but the America's Team label wasn't started by anybody associated with the Cowboys. The Cowboys were called "America's Team" long before Jones bought the team, in fact, the sihoulette of the man in your sig hated that term. And so waht if Jerry copyrigted or attempt to copyright the nickname---he's a businessman, and that name IS for better or worse associated with the Dallas Cowboys and Jerry OWNS the Cowboys.
Fact is, even with the Giants winning two Super Bowls in 5 years, it burns them up that they still take a backseat to the Cowboys in popularity. If You the NY Giants and you've won another Super Bowl, WHY are you even concerend with the Cowboys? When the Cowboys won their SUper Bowls, did the owners and GM poke fun at any other team? Not to my recollection.
Also, the Redskins have tried to claim the "America's Team' label, and when that didn't work, they came up with a knockoff that had fallen flat: they called themselves for a moment, The Nation's Team.