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 follows from the Eckmann-Hilton argument, although it is somewhat easier: just observe that the inverse is a group homomorphism $G \to G$ iff $G$ is abelian, in which case multiplication also a group homomorphism.)

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.)

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, again by the Eckmann-Hilton argument.)

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 follows from the Eckmann-Hilton argument, although it is somewhat easier: just observe that the inverse is a group homomorphism $G \to G$ iff $G$ is abelian, in which case multiplication also a group homomorphism.)