What follows is not exactly an answer but rather a gathering of a series of comments made as a reaction to Cerberus's interesting answer. See also this very recent post for a very related question, which reminded me of the older one above.

Cerberus put forward what appears to be (at least initially) a plausible proposal: the prefix of the verb effeminare has an intensive meaning ('thoroughly/completely'). My intuition is quite different. I think that the prefix here does maintain its original meaning: i.e. it does express a source. In my opinion, the (partial) paraphrase of effeminare provided by Lewis & Short is, indeed, very appropriate: 'to change out of his own nature into that of a woman'. So the root of effeminare specifies the goal (femina) but not the source. In contrast, note that the nominal root of evirare specifies the source (vir) but not the goal.

It is worth pointing out that in the case of effeminare, the root is a noun but it does not express an entity but rather a property, which makes it compatible with the pattern of prefixed verbs like efferare, emollire, exclarare, exhilarare, edulcare, etc. So in efferare-type verbs (effeminare included!) the root expresses the abstract goal (i.e. the final state), whereas the prefix points to the source (i.e. the previous state) that is not linguistically expressed. Following Lewis & Short's paraphrase of effeminare above, the meaning of efferare would be: 'to cause X to go out of his previous state into that of fierce' (where X is the direct object of the verb). See the appendix below for a relevant typological issue that provides evidence for this meaning/paraphrase.

In contrast, the formation of evirare and emasculare can be based on the productive pattern of exoculare, elinguare, edentare, where the nominal root expresses the dislocated object: i.e. 'to remove the man (understood as a property: manliness) from X', i.e 'castrate X' (X = the direct object of the verb). Alternatively, evirare and emasculare can also be formed from the pattern exemplified by prefixed verbs like erudire, where the root expresses the source state: cf. lit. 'to cause the direct object to go out of the state of rudis' with lit. 'to cause the direct object to go out of the manliness.' The pattern of the prefixed deadjectival verb erudire is that of prefixed denominal verbs like eliminare, expectorare, exterminare, etc., where the root expresses the source (note that the (dis)location pattern of these verbs is different from the dislocatum object pattern of the exoculare-type, where the root expresses the dislocated object).

A final word about the structural ambiguity in morphology is perhaps in order: perhaps some of you will be a bit puzzled by the fact that the same verb can be classified into different patterns. For example, as I said to Cerberus, exanimare is one of the verbs that has been found to be ambiguous: it can have a (dis)locatum reading (cf. also exoculare: the nominal root is the Figure) or a (dis)locatio reading (cf. also eliminare: the root is the Ground). In contrast, as predicted, the simple verb animare can only have the locatum reading (please see this link for related discussion). Of course, this kind of amibiguity is found in many languages: e.g. in English a denominal locative verb like 'to shelve' is typically used as a location verb ('to put X onto shelves') but can also be used by some speakers as a locatum verb ('to cover X with shelves').

As pointed out in my comments to Cerberus's answer, it can be useful to take a look at a Romance language like Catalan to understand a relevant typological issue involved here. The famous Talmian distinction between satellite-framed languages (e.g. Germanic languages, Slavic languages, Latin, etc.) vs. verb-framed languages (e.g. Romance languages, Turkish, Japanese, etc.) can shed some light on this issue: e.g. cf. the Path prefix (Talmy's satellite) ex- in efferare (and effeminare) with the Catalan verb sortir in Cat. 'fer sortir ferotge' (lit. 'to cause the direct object to go-out (i.e. become) fierce', where 'fierce' is the final state, i.e the abstract goal.

  The meaning of the verb-framed pattern of Cat. 'fer sortir ferotge' coincides with that of the satellite-framed pattern of ef-ferare: i.e. 'to cause someone to go out of his previous state (which is not linguistically expressed) into that of fierce'. That's why in Catalan the verb sortir lit. 'go-out' can be used with the adjective ferotge 'fierce' and the resulting meaning is not that 'one exits the state of being fierce' but rather 'one becomes fierce'. The Cat. verb sortir in 'sortir ferotge' and the Lat. prefix ex- in efferare point to the implicit source but what is expressed (morphologically in Latin and syntactically in Catalan) is the (abstract) goal, i.e the final state.

This very recent question reminded me of the older one above [note for the coordinators: I don't know if I have to answer here or there].

Cerberus put forward what appears to be -at least initially- a plausible proposal: the prefix of the verb effeminare has an intensive meaning ('thoroughly/completely'). This proposal could be said to hold for some prefixed deadjectival verbs like emollire (see below for more examples) but not for the present complex word, i.e. for effeminare. In fact, my intuition is quite different from the one expressed in Cerberus's answer. I think that the prefix here does maintain its original meaning: i.e. it does express a source. In my opinion, the (partial) paraphrase of effeminare provided by Lewis & Short at the end of this lexical entry is, indeed, very appropriate: 'to change out of his own nature into that of a woman'. Note also that the root of effeminare specifies the goal (femina).

Similarly, Michèle Fruyt, a distinguished expert on Latin word formation, has recently said the following regarding the contrast between effeminare and emasculare. Note that, following Lewis and Short above, she also advocates attributing a source state meaning to the prefix of effeminare.

le préverbe e- dans les deux lexèmes a des fonctions différentes et même opposées. Ici dans ef-femina-re, ex- renvoie à un état antérieur opposé qui a été quitté et la valeur sémantique du verbe repose sur la base du verbe. Au contraire dans emasculare, e- renvoie à l'idée d'arracher et relève du type 2 edentare 'arracher les dents'. [bold mine, Mitomino: literally, 'Here in ef-femina-re, ex- refers to an opposite previous state that has been left'.]

Fruyt, Michèle (2017). "Les verbes parasynthétiques en latin: les 2e et 3e types". De lingua latina. Revue de linguistique latine du Centre Alfred Ernout 13: 1-33.

It is also worth pointing out that in the case of effeminare, the root is a noun but it does not express an entity but rather a property, which makes it compatible with the pattern of prefixed verbs like efferare, emollire, exclarare, exhilarare, edulcare, etc. So in efferare-type verbs (effeminare included!) the root expresses the abstract goal (i.e. the final state), whereas the prefix points to the source (i.e. the previous state) that is not linguistically expressed. Following Lewis & Short's paraphrase of effeminare above, the meaning of efferare would be: 'to cause X to go out of his previous state into that of fierce' (where X is the direct object of the verb). See the appendix below for a relevant typological issue that provides evidence for this meaning/paraphrase.

In contrast, as noted above, the formation of evirare and emasculare can be based on the productive pattern of exoculare, elinguare, edentare, i.a., where the nominal root expresses the dislocated object: i.e. 'to remove the man (understood as a property: manliness) from X', i.e 'castrate X' (X = the direct object of the verb). Alternatively, evirare and emasculare can also be formed from the pattern exemplified by prefixed verbs like erudire, where the root expresses the source state: cf. lit. 'to cause the direct object to go out of the state of rudis' with lit. 'to cause the direct object to go out of the manliness.' The pattern of the prefixed deadjectival verb erudire is that of prefixed denominal verbs like eliminare, expectorare, exterminare, etc., where the root expresses the source (note that the (dis)location pattern of these verbs is different from the dislocatum object pattern of the exoculare-type, where the root expresses the dislocated object).

A final word about the structural ambiguity in morphology is perhaps in order: perhaps some of you will be a bit puzzled by the fact that the same verb can be classified into different patterns. For example, exanimare and erradicare, i.a., have been said to have a (dis)locatum reading (see the pattern of exoculare or edentare, where the nominal root is the dislocated figural object) and a (dis)locatio reading (see the pattern of eliminare, where the root is the source). In contrast, as predicted, the simple verb animare could only have the locatum reading (please see this link for related discussion). This kind of ambiguity should not be surprising since it is found in many languages: e.g. in English a denominal locative verb like 'to shelve' is typically used as a location verb ('to put X onto shelves') but can also be used by some speakers as a locatum verb ('to cover X with shelves').

As pointed out in my comments to Cerberus's answer, it can be useful to take a look at a Romance language like Catalan to understand a relevant typological issue involved here. The famous Talmian distinction between satellite-framed languages (e.g. Germanic languages, Slavic languages, Latin, etc.) vs. verb-framed languages (e.g. Romance languages, Turkish, Japanese, etc.) can shed some light on this issue: e.g. cf. the Path prefix (Talmy's satellite) ex- in efferare (and effeminare) with the Catalan verb sortir in Cat. 'fer sortir ferotge' (lit. 'to cause the direct object to go-out (i.e. become) fierce', where 'fierce' is the final state, i.e the abstract goal. The resulting meaning is not that 'one exits the state of being fierce' but rather 'one becomes fierce'.The Cat. verb sortir in '(fer) sortir ferotge' and the Lat. prefix ex- in efferare point to the implicit source but what is expressed (morphologically in Latin and syntactically in Catalan) is the (abstract) goal, i.e the final state.