Latin prefixes have no precisely defined meanings. But in any event, both sumere and assumere (= adsumere) can mean “to take up,” but on the other hand, in neither case is the idea of an upward movement particularly pronounced.
Here is the basic meaning of sumere, as defined by Lewis & Short:
to take, take up, lay hold of, assume (syn. capio)
And assumere:
to take to or with one's self, to take up, receive, adopt, accept, take
Note that even in English “take up,” the word “up” often does not refer to a discernible upward movement, and instead its meaning is also more akin to “toward the actor”; e.g. when you “take up basketball” or a sofa “takes up half the living room,” etc. The same is true for “pick up,” for example.
Likewise, what the prefix ad- in assumere confers is perhaps better described as “toward oneself,” although taking something to oneself is pretty much the meaning of plain sumere anyway, so the prefix adds little concrete meaning, and the two words are very similar. This is not unusual in Latin.
Of course, with the Assumptio Beatae Mariae Virginis, we know she goes up, because the Virgin Mary is taken to Heaven, and although I guess theologians would quickly assure us that Heaven has no physical location in our three-dimensional coordinate system, it is still widely considered to be, in whatever sense that may be, up.