Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$.
But cyclic groups of prime powers certainly do, because you'll just have $C_1\triangleleft C_p$.
Are there other cases of abelian groups with this property?