DC's Superman is best-known as the company's most popular and patriotic hero. As one of, if not the oldest superheroes ever created, Superman owes his genesis to tensions in Europe and World War II. This naturally led to Superman standing in as DC's version of Captain America, and eventually a man who would be known for "...fighting for Truth, Justice, and the America Way." But the creation of Superman's famous catchphrase was never created by his writers in the first place.
Created in 1938, Superman would soon take over his anthology book Action Comics and become one of the most popular heroes during the war period. The popular catchphrase never appeared in any comic, but rather got its start in The Adventures of Superman radio serials (and was subsequently dropped later in the war when it was clear that the Allies would win; the writers of the radio serial most likely wanted to present a unified front between the Americans, the British and the USSR). It was brought back in the 1950s television show The Adventures of Superman starring George Reeves - but again, never appeared in any comic.
In neither the radio serial nor the television show did Superman actually say the famous phrase himself. It wasn't until Richard Donner's 1978 Superman feature film that Superman, then played by Christopher Reeve, said the line (to which an incredulous Lois Lane responded "You're gonna end up fighting every elected official in this country!"). Lois wasn't the only one shocked at Superman's sincerity; the phrase at the time was more often lampooned than played seriously, especially in the aftermath of the turbulent American '60s.
In point of fact, DC Comics does not even have a current trademark on Superman's catchphrase; their last one expired in 2003, and while numerous attempts have been made by the company to trademark the phrase again, all have failed. This might be one of the reasons why Superman's new catchphrase is officially "Truth, Justice, and a Better Tomorrow", though there are certainly other reasons. As an international brand, Superman is recognized everywhere in the world - even the countries that are at odds with America. From a business standpoint, the new phrase certainly makes more sense (Americans would hardly buy something named Captain USSR in the 60s, for example).
Those who believe that Superman's catchphrase is an integral part of his identity are basing that assumption on who they believe Superman ought to be, not who he is. From a diegetic perspective, he's a hero for the entire planet, not just a single country - and from a non-diegetic view, Superman is a worldwide brand. Superman's "American Way" catchphrase is an important part of his history - but it's certainly not his most important attribute.