In Hartshorne's algebraic geometry, he said the completion of local ring $\hat{\mathscr{O}}_P(X)$ takes much more 'local properties' than the local ring $\mathscr{O}_P(X)$. There are two natural questions:
If completions $\hat{\mathscr{O}}_P(X)\simeq\hat{\mathscr{O}}_Q(Y)$, then could we deduce $\mathscr{O}_P(X)\simeq\mathscr{O}_Q(Y)$?
The example of isomorphic local rings $\mathscr{O}_P(X)\simeq\mathscr{O}_Q(Y)$, but their completions $\hat{\mathscr{O}}_P(X)$, $\hat{\mathscr{O}}_Q(Y)$ are not isomorphic? (and this happened in which situtation? maybe related to singularity?)
In particular, if the second example could be figure out, then which means that the equivalent relation 'isomorphism' is not as stronger as I imaging before in category theory.
I'll explain geometrically why the answer to your main question is a resounding no, using some basic facts about nonsingular projective curves.
Let $C$ and $C'$ be two nonsingular projective curves over $k$, and let $P, P' \in C, C'$ respectively. Then, the completions of the local rings are naturally isomorphic to $k[[t]]$ by the Cohen Structure Theorem. We will show that if $C$ and $C'$ are non-isomorphic, then there is no isomorphism between the respective local rings.
Indeed; an isomorphism of local $k$-algebras $\mathscr{O}_{C, P} \to \mathscr{O}_{C', P'}$ would (upon localizing) yield a $k$-isomorphism of function fields $K(C) \to K(C')$, which would imply that $C$ and $C'$ are birational, and hence isomorphic by Hartshorne's corollary I.6.12.
As for your second question, this cannot happen, since completion is functorial, and functors respect isomorphisms.
Edit: To elaborate on my comment, I'll give an example of how completions are useful in studying singularities. If we want to understand the singularities of curves, considering the local rings is not really sufficient. In particular, the local ring contains 'generic' data (like allowing us to recover the function field in the irreducible case) which yields a classification of singularities that is far too fine to be useful.
Completion gives a way to forget the generic data and understand only certain local properties. This gives way to a much better understanding of curve singularities. Many examples of this are in section I.5 of Hartshorne, and this is one of them, called a node.
This is the curve $\{(x + y)(x - y) = x^6 + y^6\}$, graphed in Desmos. Notice that the singularity looks a lot like the singularity obtained from the union of two lines $\{(x + y)(x - y) = 0\}$. These local rings at $(0,0)$ are clearly different since one is a domain and one is definitely not. However, you can show that both of these completions end up isomorphic to $k[[s, t]]/(st)$, so the completion is able to forget that one singularity comes from a reducible curve and the other does not.