According to Wolfram MathWorld,
"A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored ..."
and
A multiset is "A set-like object in which order is ignored, but multiplicity is explicitly significant."
In other words, both are groups of elements in which order is ignored, but whilst multiplicity is (typically) ignored in a set, multiplicity is not ignored but in fact the exact opposite—rather, it is significant—in a multiset.
But this doesn't make sense. Using the same root implies they're based off the same notion, and as multi- means some multiple of, it doesn't exactly imply some exception to that rule.
Could someone explain this to me? If B is the foundation of whatever A is, then why would B be the exact opposite of A?
The way people make up words for things is far more haphazard than you seem to believe, both in math and in the broader world. There are many examples of terms that don't really make sense when interpreted too literally. For instance, an antlion bears little direct resemblance to either an ant or a lion.
That said, I strongly disagree with your characterization of a multiset as "the exact opposite" of a set. By your criterion for "the exact opposite", any two concepts $A$ and $B$ which are not identical are "exact opposites", since you can find some property which $A$ has such that $B$ has the negation of that property. Multisets are in fact extremely similar to sets--both sets and multisets are collections of things, and in fact every set is also a multiset. They differ only in one small (but important!) way: any particular object can only be in a set one time, but it can be in a multiset multiple times. Hence the name: a multiset is something very similar to a set, except the same element can be in it multiple times.