Let $a$ be  an algebraic number, $K = \mathbb Q(a)$ the associated number field, $\mathcal O_K$ its ring of integers. Then the ring $\mathcal O_K[a]$ depends only on the set of places of $K$ at which $a$ is not integral.

Suppose there is some number field $L$ and ring extension $\mathcal O_L'$ of $\mathcal O_L$ such that $\mathcal O_K[a]= \mathcal O_L' \otimes_{\mathcal O_L} \mathcal O_K$. Then $\mathcal O_K[a]$ is a free module over $\mathcal O_L'$ and $\mathbb Z[a]$ is a finite index submodule of $\mathcal O_K[a]$. This is only a small amount of extra data beyond $\mathcal O_L[a]$ and it seems impossible to prove otherwise.

Thus let us suppose there is no such number field. Then, following YCors, suggestion, we can show that the ring of $\mathbb Z$-module endomorphisms of $\mathbb Z[a]$ is $\mathbb Z[a]$.

It suffices to show that the ring of $\mathbb Z$-module endomorphisms, tensored with $\mathbb Q$, is isomorphic to $K$. Since it contains $K$ and is contained in $M_n(\mathbb Q)$, it is a semisimple algebra, so it must be a matrix algebra of rank $r$ over a division algebra of rank $d$ over some field $L$, which commutes with $K$ and so is contained in $K$. The minimal faithful representation of such an algebra has dimension $r d^2$ over $L$, so the index of $L$ in $K$ is at most $rd^2$, and the maximal commutative subalgebra has dimension $rd$ over $L$, so the index is at least $rd^2$, so $d=1$ and the index is $r$, and the algebra is the full centralizer of $L$. In particular, if $L=K$ then the division algebra must be $K$, so it suffices to show that the set of places of $K$ at which $a$ is not integral is the pullback of a set of places from $L$.

Choose a prime $p$ over which $a$ is not integral and tensor everything with $\mathbb Z_p$. Now $\mathbb Z[a]$ is isomorphic to a finite sum of copies of $\mathbb Z_p$ and $\mathbb Q_p$ and its endomorphism ring is a ring of block-upper triangular matrices.  Tensored with $\mathbb Q_p$, this is the algebra of matrices preserving the subspace of $L$-divisible elements. Tensored with $\overline{\mathbb Q}_p$, this is the algebra of matrices preserving the intersection of the kernels of the embeddings into $\overline{\mathbb Q}_p$ corresponding to places at which $a$ is integral. This set of places is only preserved by the centralizer of $L$ if it is the pullback of a set of places fo $L$, as desired.