The Chicago White Sox took their name from an old name of the Cubs. Yes, the Cubs were, at first, known as the Chicago White Stockings.
Similarly, the Boston Red Sox took their name from an old name of the Cincinnati Red Stockings, and eventually Cincinnati dropped the Stockings and just went with Reds.
In both cases, it seems like the new team wanted to draw fans of the older name-holder.
And in both cases, Stockings got shortened to Sox, a "headline friendly" shortening.
So, it seems,
that's how these teams
got named for colored stockings.
but something less