This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring.

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.

Assume $S_1$ and $S_2$ are two finitely generated $A$-algebras (no flatness assumption), and $S_1/\mathfrak{m}^n \cong S_2/\mathfrak{m}^n$ as $A$-algebras for every $n$. If we assume $S_i$ is the inverse limit of $S_i/\mathfrak{m}^n$ respectively, then do we have $S_1 \cong S_2$ as $A$-algebras?

Again the problem is about compatibility. If one try to use the Hom functor, there is a subtlety of representability. The problem can also be stated for finite type schemes.

Moreover, I have one similar question: if $R_1$, $R_2$ are two complete noetherian local rings and $R_1/\mathfrak{m_1}^n \cong R_1/\mathfrak{m_2}^n$, then do we have $R_1 \cong R_2$ as rings?

The similar question for finitely generated modules is true as $Isom$ satisfies Mittag-Leffler condition (an endomorphism of finitely generated module is an isomorphism iff it's surjective, so this can be checked mod $\mathfrak{m}$, and Hom mod ${\mathfrak{m}}^n$ are modules of finite lengths )