Some of the teams that got their names from club names or followed that pattern:
Mutual Club (New York)
Atlantic Club (Brooklyn)
Eckford Club (Brooklyn)
Athletic Club (Philadelphia)
Allegheny Club (Pittsburgh)
Lord Baltimore Club
Maryland Club (Baltimore)
Olympic Club (Washington)
Forest City Club (Cleveland)
Kekionga Club (Fort Wayne)
Western Club (Keokuk)
When the Cincinnati Red Stockings became successful in 1869-1870, they popularized sock colors for team names—generally the socks were the only colored part of the uniform.
Some of the teams that followed this example (some of these are historically questionable):
Boston Red Stockings
Hartford Dark Blues
Worcester Brown Stockings
Worcester Grays
Troy Green Stockings
Chicago White Stockings
Louisville Grays
Providence Grays
St. Louis Brown Stockings
I think there’s a distinction to be made between the “official name” of the team and the name of the corporate entity (or entities) that owns or runs the team. The corporate entity was called the St. Louis National Baseball Club Inc., but the official name of the team has been the “St. Louis Cardinals” for a long time. The corporate entity held the rights, so you had to get the permission of the St. Louis National Baseball Club Inc. to rebroadcast portions of games that it recorded, but the team that appeared in those recordings was officially the St. Louis Cardinals.
…And the Washington American League team was officially the Nationals, incongruously; but fans and sportswriters (and the team itself, I assume) insisted on Senators and that name stuck, until Washington was vacated as a team site in 1972.
The “Nationals” name—like a lot of team names appropriated by American League clubs—is actually much older. I don’t know whether there was ever an actual “National Club,” but the name comes from the era in which teams were usually associated with gentlemen’s clubs.
The names of the Boston Red Sox, Chicago White Sox, Baltimore Orioles, St. Louis Browns, Philadelphia Athletics, Washington Nationals, Washington Senators, and Milwaukee Brewers, for example, are all based on names used by (or retroactively associated with) National League and other pre-AL clubs.
I’d quibble with this statement only to the extent that it might be in conflict with what I said above: From my perspective “New York Mets” is the actual name of the team that takes the field under the direction and sponsorship of “New York Metropolitan Baseball Club Inc.”
In 1905 the Washington American League team held a public contest to replace the nickname Senators, which apparently wasn’t very popular. The winner: Nationals, the nickname used by the 1886-1889 National League Washington franchise, which remained an unofficial nickname for years after. The Senators name emerged with the 1892-1899 National League Washington team, which originated as an American Association team in 1891.