TL/DR: Many early semi-pro and prominent amateur teams did use club names, but newspaper articles condensed those longer names into the shorter mascot-style format, which became the common format.

Most of the early professional and semi-professional teams in the US in the mid-to-late 19th century. were in baseball. Basketball hadn't been invented yet, hockey was still a long way off, and football was primarily a collegiate sport. Though football-type college games were played intramurally as far back as 1827, the first intercollegiate college football game was in 1869 (and was quite different from modern American football). The NFL was not founded until 1920 and the NBA until 1947, so aside from a few cricket squads, most of the early pro, semi-pro, and prominent amateur teams were in baseball.

Baseball was popular in the cities, so it followed that there were many different teams within the same city and thus calling the team just by the city name would not suffice. It's okay if you're the only team in the city and can call yourself the New York Base Ball Club, but as baseball began to eclipse cricket in popularity in the late 1850s and then exploded after the Civil War, a nickname of some sort was necessary. Many of these early teams did use club-style names. The first boxscore in a newspaper detailed a game between the New York Club and the Brooklyn Club (New York Morning News, October 22, 1845). The New York Club was the New York Knickerbockers, who were then known as the Knickerbocker Base Ball Club of New York. The New York Clipper of August 9, 1856, reported on a game between the Union Club of Morrisania (New York) and the New York Baltics. One early game often cited in histories of baseball statistics and reporting was played in 1858 between Resolute (Brooklyn) and the Niagara Club (South Brooklyn); that one was notable for being believed to have the first reported pitch count. The starters for each side, R.S. Canfield for Resolute and John Shields for Niagara, combined for 812 pitches, which sounds unbelievable until you remember that back then pitchers threw underhanded and were there more to get batters to put the ball in play than to try to strike them out.

One of the earliest successful amateur teams was the Excelsior Club of Brooklyn, which featured Jim Creighton, perhaps baseball's first star player, and future Hall of Famer Candy Cummings, credited with inventing the curveball. Often newspaper accounts of the team's exploits would condense the wordier "Excelsior Club of Brooklyn" into the shorter "Excelsiors." The same thing happened with another prominent team of that city, where newspaper accounts shortened "Atlantic Base Ball Club of Brooklyn" to "Atlantics." Later on it would become common to prepend the city name, hence Brooklyn Atlantics and Brooklyn Excelsiors. One example of this still survives: an 1860s team called Athletic of Philadelphia became the namesake for a charter member of the American League in 1901, the Philadelphia Athletics, later the Kansas City Athletics and today the Oakland Athletics. So once it became common for teams to be referred to by plural nouns, that became the typical way new teams called themselves. By the time the National League was founded in 1876, three of the eight teams were nicknamed for colors (Hartford Dark Blues, Cincinnati Reds, Louisville Grays) and three for colored hosiery (Boston Red Stockings, Chicago White Stockings, St. Louis Brown Stockings).