The Cincinnati Reds name is derived from hosiery. In 1869 the first openly-professional (not that there's anything wrong with that) baseball team was the Cincinnati Red Stockings. They went undefeated for their first year (57-0) and a half (81-0 overall), from their first game in April 1869 until June 1870. After their first loss, interest in the team quickly waned. Fair weather fans in the extreme. They were disbanded in 1871, and some of the players regrouped in Boston to form the Boston Red Stockings. The Boston team eventually became the Atlanta Braves.
In 1876 the new National League was formed and one of the teams was the Cincinnati Red Stockings, reclaiming their old name. Four years later in 1880 the National League expelled the Red Stockings from the league for selling beer at the games (forbidden at ballparks in the NL) and for playing games on Sundays. (the Cincinnati Reds (in various flavors of the name) were the first team to sell beer at games, the first to play on Sunday, the first to play a night game under artificial lighting, and the first to have a game televised).
Two years later the American Association was formed with the Cincinnati Red Stockings in charge, and teams were located in "river towns" like Pittsburgh, Cincinnati, Louisville and St. Louis, where the cities were populated by low-class citizens. The league had cheaper prices than the National League, and they sold beer and whiskey at the games. The American Association, or AA, was known as the "Beer and Whiskey League," and isn't that funny. In 1890 the Cincinnati Red Stockings, as the Cincinnati Reds, were re-admitted to the National League, Sunday games and beer in tow.
During the McCarthy Era, thanks to the Red Menace and the "Second Red Scare," the Reds changed their name to Redlegs. I don't know what this fetish with hosiery is all about in Cincinnati.
The Boston Red Stockings later changed their name to the Boston Red Caps, in deference to the newly re-created Cincinnati Red Stockings. Then they became the Boston Beaneaters. Over the next few years they had a flurry of name changes, including the Doves, Rustlers and finally the Braves when James E. Gaffney bought the team Gaffney was a product of Tammany Hall, the NYC political machine. Tammany was named after Native American Chief Tamanend of the Lenni-Lenape Indian Nation of the Delaware Valley. Tamanend and later the Tammany Society at Tammany Hall used an image of an Indian Brave as their iconic signature. Ironically, Chief Tamanend was best known for his peaceful nature and peaceful politics of negotiation. The Boston Braves later moved to Milwaukee and then Atlanta where the tomahawk reigns supreme in peaceful negotiation venue.
The Washington Redskins were originally the Boston Braves, the same name as the baseball team who owned the stadium in which the football team played their games. When the team moved their games to Fenway Park, where the Red Sox played their games, the football team's name was changed to Redskins, because Redskin was kind of like an Indian Brave but wasn't the same as the "Braves," and the football team's owner didn't want to name his team after hosiery or head wear.