What does league, meaning “alliance,” have to do with league, meaning “a few miles”? An attempt to find kinship between these words by positing the notion of linking several similar entities or units is futile: The identical formation of the words is coincidental.
The first sense of league stems from the Latin verb ligare, meaning “bind” and the basis of ligament, meaning “band” or “bond,” and ligature, which means, among other things, “something that binds or connects.” League originally referred to a confederacy of geopolitical units (such as the Achaean League of classical Greece and the Hanseatic League, an economic alliance established in the early Middle Ages, as well as the League of Nations of the early twentieth century) but later came to apply as well to political associations and athletic organizations.
People or organizations that conspire are said to be in league with each other, and when someone outclasses someone else in terms of some characteristic, the first person is said to be out of the other person’s league, while “in a league of (one’s) own” means “superior skill or status.” By contrast, “in the same league” means “of comparative skill or status.”
On a related note, the expression “Ivy League,” from the name of the collegiate athletic league populated by eight of the nation’s most prestigious universities, by extension denotes the schools themselves as well as high social and cultural status and elitism. (The reference to ivy pertains to the walls of venerable school buildings being covered in ivy over the years.)
League is also a verb meaning “unite,” but the verb beleaguer, meaning “besiege,” is unrelated. Idioms pertaining to the “confederation” sense of league include “major league,” originating in the term for the highest level of professional baseball but by extension alluding to significant actors or entities in a realm of human endeavor (“in the big leagues” has the same sense); “minor league,” denoting something of inferior status (from the lower caliber of play in baseball’s minor leagues); and “bush league,” which, based on the slang term for semiprofessional baseball (from the expression “the bush,” referring to a rural area) suggests petty, unprofessional behavior. (The last term was not always pejorative, however.)
The sense of league of a measure of distance derives from the Latin noun leuga and is primarily understood to refer to a distance of three miles, though it has applied to measures ranging from about two and a half to approximately four and a half miles. (It can also apply to a square measuring about three miles on a side.) A league, thought to originate as the distance traveled on foot in one hour, it is no longer an official distance.
The French term banlieue, meaning “suburb” but increasingly connoting low-income housing projects (though banlieues diverge widely in economic status), is a geopolitical term, but it is distantly related to the latter sense of league: It is a compound ultimately derived from the Germanic terms ban, meaning “proclamation,” and leuca, meaning “league,” with the connotation of “area outside the city but within its legal jurisdiction.”