The first three Giants-Yankees matchups were in 1921, 1922, and 1923. However, in 1921 an 1922, the teams played in the same stadium at the Polo Grounds. The series was nicknamed the "Battle of Coogan's Bluff." The 1923 matchup was called an "all-New York World Series."
"Subway Series" was used at the 1934 all-star game that featured many Giant and Yankee players. However, the New York teams did not both make it to the World Series that year. The first "Subway Series" was actually the fourth, in 1936.
The subway fare used to be a nickel, and the term "Nickel Series" (first cited in 1927) appears earlier than "Subway Series" (first cited in 1928).
Since 1997, the term "Subway Series" has also applied to interleague play during the regular season. However, the blog Uncle Mike's Musings: A Yankees Blog and More insists that "Subway Series" is strictly for a World Series, and that interleague play during the regular season is a "City Series."
When the New York Giants played the Brooklyn Bridegrooms (they would be called the "Trolley Dodgers" in 1895) for the 1889 World Series, there was no subway. The New York City subway opened in 1904. Some have dubbed the 1889 World Series the "Trolley Series." This is a poor name, however, because the 1944 World Series between the St. Louis Browns and the St. Louis Cardinals is usually known as the "Trolley Series."
Wikipedia: Subway Series
The Subway Series is a series of Major League Baseball (MLB) rivalry games played between teams based in New York City; every historic and current venue for such games has been accessible via the New York City Subway, hence the name of the series.
The term's historic usage has been in reference to World Series games played exclusively between the city's teams. The New York Yankees have appeared in all Subway Series games as they have been the only American League (AL) team based in the city, and have compiled an 11–3 all-time series record in the 14 championship Subway Series.
Since 1997, the term Subway Series has been applied to interleague play during the regular season between the Yankees and New York City's National League (NL) team: the New York Mets. The Mets and Yankees also played each other in the 2000 World Series.
4 August 1927, Atlanta (GA) Constitution, pg. A3:
Most New Yorkers would rather have all the games of the 1927 world series within a nickel's ride of their homes.
25 September 1928, Bee (Danville, Virginia), pg. 3, col. 3:
I feel that I am violating no confidence in saying that it looks like New York in the American League. The other half of the proposed subway series may or may not be closer to its goal line by tonight but no one can say that their chances are only nominal.