In a category of modules (or more generally in any additive category), every object has a unique group object structure. Indeed, a group object structure on a module $M$ consists of just homomorphisms $e:0\to M$, $i:M\to M$, and $\mu:M\times M\to M$ such that the binary operation $\mu$ satisfies the group axioms with $e(0)=0$ as its unit and $i$ as the inverse map. The fact that $\mu$ is a homomorphism forces it to just be the module addition operation in $M$: $$\mu(a,b)=\mu(0+a,b+0)=\mu(0,b)+\mu(a,0)=b+a$$ (this trick is known as the "Eckmann-Hilton argument"). So the only possible group object structure is the one given by $\mu(a,b)=a+b$ and $i(a)=-a$. Conversely, since these maps are indeed homomorphisms, they do give any object $M$ a group structure.
For Lie algebras, we can use the exact same argument to show that any Lie algebra $A$ has at most one group object structure, given by $\mu(a,b)=a+b$ and $i(a)=-a$. However, in this case, these maps usually are not actually homomorphisms at all. Indeed, in order for $\mu$ to be a homomorphism, we must have $$[a,b]=\mu([a,b],0)=\mu([a,b],[a,a])=[\mu(a,a),\mu(b,a)]=[2a,b+a]=2[a,b]$$ so $[a,b]=0$ for all $a,b\in A$ and $A$ is abelian. Conversely, if $A$ is abelian, then $\mu$ and $i$ are indeed homomorphisms. So, a Lie algebra has a group object structure iff it is abelian, and the group object structure is unique.