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?