TLDR: existence yes, uniqueness no.
Arun has already discussed existence, so let me discuss the uniqueness part of the question. It is a special case of whether cancellation holds in bordism categories, which it usually doesn't except under rather strong assumptions.
The uniqueness part of the question is equivalent to whether all embeddings $\Sigma_{M_1,M_2} \hookrightarrow \Sigma$ relative to $M_1 \amalg M_2$ are in the same orbit under the action of the group of diffeomorphisms of $\Sigma$ restricting to the identity on $\partial \Sigma$. (One direction: given two such embeddings, their complements would be two candidates for $\Sigma'$ and any diffeomorphism rel boundary between these two complements may be glued to the identity map of $\Sigma_{M_1,M_2}$ to get a diffeomorphism of $\Sigma$ which acts one embedding into the other. The converse is similar.) In fact we may construct an example of two embeddings $e, e': \Sigma_{M_1,M_2} \to \Sigma$ where it is not even possible to find a continuous map $f: \Sigma \to \Sigma$ which is the identity on $\partial \Sigma$ and such that $f \circ e \simeq e'$ relative to $\partial \Sigma$.
To build a counterexample we use that for any group $G$, the set of $G \times G^\mathrm{op}$-equivariant maps $G \to G$ (action by multiplying on both sides) is precisely the maps given by multiplication by elements of the center of $G$. Hence if $Z(G) = \{1\}$ it consists of only the identity. This implies the following more complicated looking (but "isomorphic") statement: if $C$ is a groupoid and $m_1$ and $m_2$ are two isomorphic objects in $C$ such that the group $\mathrm{End}_C(m_1)$ has trivial center, then any functor $F: C \to C$ with $F(m_1) = m_1$ and $f(m_2) = m_2$, and $F(f) = f$ for any $f \in \mathrm{End}_C(m_1)$ and any $f \in \mathrm{End}_C(m_2)$, also has $F(f) = f$ for any $f \in \mathrm{Hom}_C(m_1,m_2)$.
Now pick a finitely presented non-trivial group $G$ with trivial center and build a connected oriented cobordism $W$ from $M_1 \amalg M_2$ to any connected $N$, such that there exists an isomorphism $\pi_1(W,w) \cong G$ for some, hence any, $w \in W$, and such that both inclusions $M_1 \to W$ and $M_2 \to W$ induce isomorphisms in fundamental groups. For example, we could first pick a closed connected oriented manifold $M$ with the right fundamental group (this is possible in any dimension $d \geq 4$), then set $M_2 = M_1 = M$ (with opposite orientations), and let $\Sigma$ be the complement of a ball in the interior of $[0,1] \times M$, regarded as a bordism from $M_1 \amalg M_2$ to $N = S^{d-1}$.
Then pick points $m_1 \in M_1$ and $m_2 \in M_2$, which we can think of as objects in the fundamental groupoid $\pi_1(\Sigma)$. Connectedness implies that these objects are isomorphic, and we have arranged that the group of endomorphisms of $m_1$ has trivial center. By what we saw above, any functor $F: \pi_1(\Sigma) \to \pi_1(\Sigma)$ which sends both objects $m_1$ and $m_2$ to themselves and act as the identity on their endomorphism groups, must also act as the identity on the set of isomorphisms from $m_1$ to $m_2$.
The bordism $\Sigma_{M_1,M_2}$ may be constructed by attaching a 1-handle to $M_1 \amalg M_2$ at the points $m_1$ and $m_2$. Then any embedding $\Sigma_{M_1,M_2} \to \Sigma$ gives rise to an isomorphism (namely the path through the attached 1-handle) from $m_1$ to $m_2$ in the fundamental groupoid of $\Sigma$, and any element of this Hom-set arises in this way (since any path may be perturbed to a smooth embedding, which by the orientability assumptions may be thickened up to an embedding of the handle). Then pick two embeddings $e, e': \Sigma_{M_1,M_2} \to \Sigma$ representing distinct elements in the Hom-set. Any diffeomorphism of $\Sigma$ relative to $M_1 \amalg M_2$ which acts $e$ into $e'$ would induce a functor $\pi_1(\Sigma) \to \pi_1(\Sigma)$ acting as the identity on the endomorphisms of $m_1$ and $m_2$ (since these endomorphism groups may be calculated in $M_1$ and $M_2$) but not on the set of morphisms between them. But that's precisely what we ruled out! Hence such a diffeomorphism cannot exist.