As typically with this kind of thing, it's good to look at the different meanings present already in classical Latin.
Lewis and Short mention "collection, gathering" as a meaning of conceptus.
And that makes sense: capere means taking or capturing, con- adds togetherness.
Taking things together is gathering.
But that is not the only meaning.
It is not unusual to use con- as an intensifier — and indeed it is quite common in Latin to use various prefixes as intensifiers disregarding their literal meaning.
One of the meanings of conceptus is indeed "gathering", but that seems to be quite unrelated to what you are after.
It is useful to dig a little further within Latin.
The word conceptus is derived from concipere.
Broadly, its meanings are variations on the theme of capere; see the linked L&S entry for more details.
One sense of capturing this verb is used for is understanding.
And it's not that far fetched: to grasp something mentally is much like taking hold of something physically, and this is reflected in many languages.
The meaning of conceptus you are asking about seems to be related to this meaning (II in L&S) of concipere.
Based on this, I would answer your questions as follows:
Does the prefix con- truly mean “with” here? But Etymonline says that it's "probably an intensive prefix".
No, I would not interpret con- as "with".
Literal interpretation of Latin prefixes is often ill-advised, in part due to words changing meanings unpredictably in time.
It is indeed probably mostly about intensity.
If con- does mean “with”, then how did “with-taking” semantically shift to mean "(a thing) conceived"?
The prefix is mostly an intensifier used to tone the verb capere, "to catch".
Concipere is used in the mental sense (among other senses), so that concipere can mean "to grasp", "to think of something".
From this verb one can then derive the fourth declension noun conceptus, "the process of concipere" or "conception".
One can also derive the second and first declension participle conceptus, "the [thing/person/other] that has been grasped".
This is very literally "a thing conceived", so the description you cite is quite accurate in that sense.
I hope it makes more sense now!