Here's my initial takeaway:
According to the Wiktionary links you included, evirare is a contraction of ex-virare, so something like "to take out of a man". In Greek (maybe sometimes in Latin too), this use of ex- (Greek εξ-) would take the genitive as a genitive of separation (as in "he came out of the woods, he walked out of the house" in English). From what I've encountered, this use is usually soaked up by the ablative case in Latin which Greek does not have.
Neither in Wiktionary nor in Lewis and Short is effeminare described as a contraction with ex-. For now, I assume it's just a simple verb-ization of feminare with the e maybe a vocal affect (I'm thinking of Spanish "espiritu" from spiritu, or "estar" from stare).
If we do take effeminare to be a contraction with ex-, it would be comparable to the Greek genitive of accompaniment, which, despite using the same case and form, does hold the opposite meaning. We do this in modern English too: "out of curiosity" would probably be more sensible as "in curiosity", "out of sheer force of will" as "in sheer force of will". Meanwhile, the word "out" holds perhaps an opposite meaning in phrases such as "out of line" or "out of order". Particularly in the second phrase, "out" holds the opposite meaning to "in", but "out of curiosity" and "in curiosity" are synonyms.
The other classical language example I can think of where a prefix/preposition can hold opposite meanings is the distinction between the Greek alpha privative and the copulative alpha. Both consist of the prefix α-, but while the alpha privative negates the following word (think modern English "atypical", "asexual", "apolitical"), the copulative expresses sameness as in the Greek word for brother, αδελφος, meaning literally "of the same womb".
In summary, I don't think effeminare is a contraction with ex-. But if it is, that's okay, because ex can have either an inclusive or an exclusive meaning.