For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$.
As far as I understand, for non-orientable manifolds connected sum is not quite well-defined, at least it can be non-unique (or is the problem worse than that?). Suppose $M\#N$ is one of possible realizations of connected sum (even if other non-homeomorphic realizations exist).
Does the above equality still holds for it?
If not, then: even if the manifolds decomposable as $M\#N$ can differ, don't they have the same fundamental groups?