Interestingly this random "the" actually appears twice in Japan. There is "the Ginza" in Tokyo, today the height of cosmopolitan retail, and "the Gion" in Kyoto, the traditional street of geisha and teahouses (and tourists). But Japanese is a notoriously vague language and doesn't actually posses a definitive article so why do we sometimes impose one in English? Indeed, in Japanese the name is sometimes reverse engineered through katakana (the alphabet reserved for western words) as "za Ginza", a confirmation, if needed, that the origins are English, rather than Japanese.
The etymology of the name itself relates to the government's silver mint (Ginza literally translates as "Silver Chair") which was located in the area until 1800 when the shogunate became tired of the endemic corruption and moved it to Nihonbashi where they could keep a closer eye on back door activities. The name stuck though and the area was a veritable rabbit warren of kabuki theatres, river boat wharfs and kimono stores until in 1872, it was razed by catastrophic fire. And now it began to take a very different shape. An Irishman, by the name of Thomas James Waters, cleared the streets and created the new and distinct European coffee house experience of Brick Town.
English speaking foreigners were just beginning to appear around this time in Japan as the country opened it's doors to the world and it was during this period that it seems to have acquired the "the". Reference to it can be found as far back as 1908 in The New York Times when it was reported that "Admirally Sperry was mobbed by crowds wanting to shake his hand in the Ginza". And then again in the Chicago Tribune where somewhat more ominously the front page reported "today's target is the Ginza" in January 1945. And so the Ginza's epithet arose as a result of its position as a unique European experience in the heart of the capital city of the land of the rising sun. Somewhere to meet and discuss the events of the day. "The" place to be seen.