This was generally considered a superstition rather than any kind of hard science or deep-seated religious belief. Pliny the Elder mocked it in his Natural History from 79 AD: “The falsehoods of the magicians would persuade us that these stones are preventive of inebriety, and that it is from this that they have derived their name.”

There are two possible sources of someone ending up believing in this myth:

1. Many aspects of drunkenness are mental rather than physical. Blind studies involving unknowing participants in a setting designed to encourage the feel of a bar found that lowered inhibitions and even a flushed face can occur in sober people who believe they're being given alcohol. Someone with an amethyst cup who believed they were being kept sober could have legitimately felt that way even if they were drinking.
2. The Greeks and Romans both watered down their wine, considering the drinking of unmixed wine barbaric. Even in the most debaucherous parties, wine was watered down in a 1:1 or 1:2 ratio. In daily life or when engaging in more serious philosophical discussion, 1:3 or even 1:4 was the norm. I've tested this myself many times and an entire pint glass full of 1:3 wine-to-water mixture looks exactly like regular red wine but tastes like water with a faint wine aftertaste and only has the alcohol content of a standard 6 oz. glass of undiluted wine. A typical Greek or Roman dinner guest could drink an entire gallon of 1:3 mixture and have had the equivalent of 7 glasses, enough to be drunk but not uncontrollable depending on their tolerance.