"Betäubung" has a similar etymology as "narcotic". Both mean to numb the senses or put to sleep (hence e.g. "narcolepsy"), and in German it's therefore also used for sedatives and anesthetic drugs. In modern use, "narcotic" has also semantically shifted to include any illegal drug, as with "Betäubungsmiddel".

Interestingly, in both cases the semantic shift seems to have been caused by the enactment of laws to control drugs. The legal term these days is probably "controlled substance" in English, but "narcotic" now definitely refers to many drugs that are not medically narcotic.

It can also mean anesthetics, which coincidentally would include cocaine as a strong local anesthetic, but not a narcotic in the pharmacological sense.