If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $_\infty X$ of $_n X$ to $X$ is an isomorphism, where $_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positive integers $n$ ordered by divisibility. (The only thing to check is stability under extensions of this class of objects; thanks to AB5 the proof is the same as for abelian groups.) If $M$ is such that $n.Id_M$ is a mono for each integer $n>0$ (it is true for any subobject of a uniquely divisible object), then $M\to\, _n X$ is always zero, so $M\to\,_\infty X$ is always zero because $A$ is AB5.

AB4 is not enough: we have an epimorphism of abelian groups from the uniquely divisible group $\mathbb{Q}$ onto the nonzero torsion group $\mathbb{Q}/\mathbb{Z}$, which belongs to the smallest Serre subcategory of abelian groups stable under products. By dualising one sees that the opposite category of abelian groups does not satisfy the property.

If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $_\infty X$ of $_n X$ to $X$ is an isomorphism, where $_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positive integers $n$ ordered by divisibility. (The only thing to check is stability under extensions of this class of objects; thanks to AB5 the proof is the same as for abelian groups.) If $M$ is such that $n.Id_M$ is a mono for each integer $n>0$ (it is true for any subobject of a uniquely divisible object), then $M\to\, _n X$ is always zero, so $M\to\,_\infty X$ is always zero because $A$ is AB5.

I guess that AB4 is not enough.

If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $_\infty X$ of $_n X$ to $X$ is an isomorphism, where $_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positive integers $n$ ordered by divisibility. (The only thing to check is stability under extensions of this class of objects; thanks to AB5 the proof is the same as for abelian groups.) If $M$ is such that $n.Id_M$ is a mono for each integer $n>0$ (it is true for any subobject of a uniquely divisible object), then $M\to\, _n X$ is always zero, so $M\to\,_\infty X$ is always zero because $A$ is AB5.

AB4 is not enough: for any prime $p$ the additive group of $p$-adic integers, which belongs to the subcategory of abelian groups which is generated by torsion ones and is stable under products, imbeds into the uniquely-divisible abelian group of $p$-adic rationals. So, dualise and look at the opposite category of abelian groups.

If your category $A$ is AB5, the answer is positive, because the ind-torsion objects are only the objects $X$ of $A$ such that the canonical map from the colimit $_\infty X$ of $_n X$ to $X$ is an isomorphism, where $_n X$ is the kernel of multiplication by $n$ on $X$ and the (filtered) colimit is taken over positive integers $n$ ordered by divisibility. (The only thing to check is stability under extensions of this class of objects; thanks to AB5 the proof is the same as for abelian groups.) If $M$ is such that $n.Id_M$ is a mono for each integer $n>0$ (it is true for any subobject of a uniquely divisible object), then $M\to\, _n X$ is always zero, so $M\to\,_\infty X$ is always zero because $A$ is AB5.

AB4 is not enough: we have an epimorphism of abelian groups from the uniquely divisible group $\mathbb{Q}$ onto the nonzero torsion group $\mathbb{Q}/\mathbb{Z}$, which belongs to the smallest Serre subcategory of abelian groups stable under products. By dualising one sees that the opposite category of abelian groups does not satisfy the property.