Why isn't $BG$ a group, for $G$ not abelian? If $G$ is a discrete or topological group, $G$ is a closed subgroup of $EG$, and normal iff $G$ is abelian, according to Segal, Cohomology of topological groups, Symposia Mathematica IV (1970) (a reference I found from two answers by Chris Schommer-Pries: Classifying Space of a Group Extension and Good functorial model for BG). Hence for $G$ not abelian, $G$ is not normal in $EG$, hence $BG=EG/G$ is not a group.
On the other hand, we also learn from Segal that the reason $EG$ is a group is that the universal bundle functor $E$ is monoidal with respect to the Cartesian monoidal structure. Any monoidal functor takes group objects in one category to group objects in another simply by functoriality. But the classifying space functor $B$ is also monoidal (see Good functorial model for BG or Peter May's answer at Group structure on Eilenberg-MacLane spaces). Hence $BG$ should be a group?
 A: The classifying space functor may be a monoidal functor out of $\text{Grp}$, but nonabelian groups aren't group objects in $\text{Grp}$. (The group objects in $\text{Grp}$ are precisely the abelian groups. This is also a corollary of the Eckmann-Hilton argument.)
A: A simple answer for why $BG$ is not a group when $G$ is discrete is that $BG=K(G,1)$, so $\pi_1(BG)=G$. However, if $K(G,1)$ were a topological group, then $\pi_1(K(G,1))=G$ must be abelian. This is proved in the usual way: the pointwise product of two loops in the group structure on $BG$ is homotopic to the concatenation in either order, by homotoping the first loop to be the identity for the first half of the interval, and homotoping the second loop to be the identity for the second half of the interval, or vice-versa. 
A: [As a non-homotopy theorist, I am not certain about this answer.]
If your topological group $G$ has a $2$-fold loop space structure, but not an infinite loop space structure, then $BG$ is naturally a group, but there is some $n \geq 2$ for which $B^nG$ is not.  In this case, $G$ is not strictly abelian, despite $BG$ being a group.  In fact, even in the infinite loop space case, you can have groups that are not strictly abelian, like the infinite unitary group $U$.
As other answers have pointed out, this behavior is not possible when $G$ is discrete.  Indeed, 2-fold loop structure automatically strictifies to the abelian property in the discrete case.
