Without specifying the drink then yeah it really implies alcohol (just like the English "don't drink so much"). It doesn't have to mean that but that's probably going to be the first association that pops into someone's mind.
I'd say it's not incorrect, but sounds much clumsier than не пей так много. You can also say Так много не пей. It depends on what you're emphasising.
You can say "Не так много!" If both of you know that it i is about drink or if it is in the moment of drinking.
In your translation, "that" is a pronoun. "Так", however, is an adverb, so "that" is only a possible translation when used as an adverb (as in the official translation).
don't drink that much or don't drink too much but don't drink so much ... no no no
"Don't drink so much" is correct. "That much" is also correct. "Too much" is wrong, it would be "слишком много" not "так много".