I have several times heard anecdotes on the theme that the words we use for genitals today were considered very vulgar back in Rome during ancient times, similar to cock/cunt etc in modern vocabulary (and the "punchline" with these anecdotes were along the lines that "your feeling of sophistication when you use Latin (derived) words today would be laughable for the Romans").
What words did the educated as well as non-educated Romans use when referring to (human) genitals in formal settings?
It doesn't quite work as your source has stated. The "proper" terms are mentula and cunnus, which are obscene but not slang, i.e. mentula and cunnus first and primarily referred to male and female genitalia respectively.
In polite speech, the Romans employed a variety of different euphemisms, including penis and vagina, but it should be noted that they often endeavored to avoid the discussion altogether. Formal works (such as orations and epics) were reticent about discussing sex and especially genitalia.
When they did, they often did not distinguish between the sexes, and in fact genitalia comes directly from the Latin and refers, like it does in English, to either sex organs. Other words that accomplish the same are veretrum, verenda, pudenda, some phrase with natura, and even sexus.
For specifically the male membrum, in fact membrum (especially membrum virile) was used, as well as particula ("the little part"), while for the female genitals might be metaphorically described with words for fields or caves. More euphemistically, you see sinus muliebris as a good equivalent to membrum, as well as partes muliebres and even once muliebre membrum, although that comes down from a very late source (Ausonius, 4th century CE).
There are, of course, many other euphemisms, both more formal and some that were slang, that were employed. You will not find a single word across all authors that otherwise designates a "formal" word for the male and female genitals.
One additional note: penis was originally just a euphemism, but did indeed become more obscene over time (as early as Cicero!). Vagina too was just a sex joke in Plautus. It's less "dick/cunt" though and more "shaft/hole".
If you're interested in this topic in more detail, I encourage you to read Adams' 1982 monograph The Latin Sexual Vocabulary.
