I know that some Association Football clubs do have nicknames, but you usually don't hear them. The wiki page tells me that Manchester United are nicknamed the "Red Devils" but that mascot wasn't adopted until the 1960s, and you rarely see a "Manchester United Red Devils" headline. Whereas, North American teams like the New York Giants and Boston Red Sox have been referred to by their mascot name since at least the early 1900s.
Has this always been the case? Why do clubs outside North America, particularly football/soccer clubs, not use team nicknames to the same degree as they are used in baseball, basketball, hockey, and American and Canadian football? Or put another way, why are team nicknames used so extensively in North America unlike elsewhere?
New York Yankees is not the team's nickname it is their official team name. The Yankees have nicknames such as The Bronx Bombers, but that would not be used in an official capacity. However, you could see the team in a score book referred to as either New York Yankees, Yankees or New York (although New York has two teams so you may add AL for the American League).
Here are a couple teams that are not actually official team names. The Montreal Canadiens or Habs are officially called "le Club de hockey Canadien". Inter Milan's official name is "F.C. Internazionale Milano".
For team mascots, it is the Giants have 'Lou Seal' and the Red Sox have 'Wally the Green Monster' while the Yankees are too good for a mascot.
Back to it, there's a lot of leagues and teams, each with separate histories although influenced by others. Newer teams will generally follow an accepted pattern, while older teams and leagues may not. I think it will be tough to answer generally why one league went one direction and another did it differently.
Some helpful graphs to read more name origins are http://www.flipflopflyin.com/flipflopflyball/info-teamnames.html https://en.wikipedia.org/wiki/Association_football_club_names