Let $M$ and $N$ be two smooth manifolds with $$\textrm{dim}(M)=\textrm{dim}(N)=n.$$ Let $U\subseteq M$ and $V\subseteq M$ be two open sets and $f:U\longrightarrow V$ a smooth diffeomorphism. Consider the equivalence relation in the disjoint union $M\sqcup N$: $$x\sim \phi(x), \forall x\in U.$$ Consider $M\cup_f N:=(M\sqcup N)/\sim $ with the quotient topology.
Is it true that there is a unique smooth structure on $M\cup_f N$ such that the inclusions $\imath:M\longrightarrow M\cup_f N$ and $\jmath:N\hookrightarrow M\cup_f N$ are embeddings?
Since your comment made clear that your definition of manifold includes the Hausdorff condition, the answer is no: in general, $M\cup_f N$ will not even be a topological manifold. A simple counterexample is obtained by taking $M=N=\mathbb R$, $U=V=\mathbb R\smallsetminus \{0\}$, and $f\colon U\to V$ to be the identity map. Then $M\cup_f N$ is the "line with two origins," which is not Hausdorff.