Quotient of the direct product of cyclic groups
Solution 1:
Summarizing the comments; it is not true in general. The simplest example is the quotient $$(\Bbb{Z}_4\times\Bbb{Z}_2)/\Bbb{Z}_2.$$ Then the isomorphism type of the quotient depends on whether you take the quotient w.r.t. the subgroup $\Bbb{Z}_2\subset\Bbb{Z}_4$ of the first factor, or the subgroup $\Bbb{Z}_2\subset\Bbb{Z}_2$ of the second factor. And there is even a third subgroup $\Bbb{Z}_2\subset\Bbb{Z}_4\times\Bbb{Z}_2$, that is not contained in either factor.