Sock color was your distinctive uniform feature as most uniforms were white or gray... Original Cincy Red Stockings folded in 1870, George and Harry Wright and other players founded Boston team with same name 1871. Cincy rebooted it's team in 1876 and joined NL, also referencing back to 1869 team.
Had to be confusing to read something like "the Red Stockings throttled the Red Stockings 8-1."
Nicknames were much more fluid (and descriptive) and usually only used when it made sense. They would most likely have said the Bostons throttled the Cincinnatis... Official names would have been more like the Boston National League Base Ball Club and the Cincinnati National League Base Ball Club...
That makes sense.