Actually, as you may know, there was long a AAA Milwaukee Brewers team. From 1902-52 they played in the American Association; so as soon as that Western League franchise left town, a new ballclub formed (also courtesy Wikipedia).
It may have occurred to the Braves to change their nickname to Brewers when they arrived, but for some reason they resisted the temptation. (Most teams keep their old nicknames in their new town, such as the Utah Jazz). But with Milwaukee's breweries, Brewers is rather inevitable.
They got the name because they played at Fenway, home of the Red Sox. At the time they were coached by a guy who billed himself as an Indian, although it later turned out not so much. So, it was a "Red" thing.