Drinking might've been underground, but speakeasy patrons wouldn't take just any cocktail
The practice of ordering brand name booze actually started during Prohibition, since people were wary about what was in the unregulated well liquor the bootleggers supplied. They were also too insecure to just ask for a Long Island Iced Tea.
It wasn't the FBI running the no-booze enforcement
It was the IRS. Sure, they called it the Bureau of Internal Revenue then, but yep, the tax geeks were running point. Any and all "agents" were part of the Bureau's Prohibition Unit, which eventually grew so large that it spun off and became its own bureau in the Department of Treasury in 1927, and later was adopted by the Department of Justice in 1929. Still, the numbers men were the bosses throughout all the shuffling. Even Eliot Ness and his Untouchables crew were revenue guys, not G-men. (In fact, the FBI turned Ness down when he applied in 1933. Scandalous.)