the Azumaya locus does determine the order

Clearly
this cannot be correct for consider for $n\in \mathbb{N}$ the order

$A_n=\left[\begin{array}{cc}\mathbb{C}[x]& \mathbb{C}[x]\\ (x^n)& \mathbb{C}[x]\end{array}\right]$

For $m\ne n$ the orders $A_n$
and $A_m$ have isomorphic Azumaya locus, but are not isomorphic as
orders. Still, the statement in the heading is _morally_ what Nikolaus
Vonessen and Zinovy
Reichstein are proving in their paper Polynomial identity
rings as rings of functions. So I better clarify what they do claim
precisely.

Let $A$ be a _Cayley-Hamilton order_, that is, a
prime affine $\mathbb{C}$-algebra, finite as a module over its center
and satisfying all trace relations holding in $M_n(\mathbb{C})$. If $A$
is generated by $m$ elements, then its _representation variety_
${\mathbf{rep}}_{n}A$ has as points the m-tuples of $n\times n$ matrices

$(X_1,\dots ,X_m)\in M_n(\mathbb{C})\oplus \dots \oplus M_n(\mathbb{C})$

which satisfy all the defining relations of
A. ${\mathbf{rep}}_{n}A$ is an affine variety with a $G{L}_n$-action
(induced by simultaneous conjugation in m-tuples of matrices) and has
as a Zariski open subset the tuples $(X_1,\dots ,X_m)\in {\mathbf{rep}}_{n}A$ having the property that they generate the whole
matrix-algebra $M_n(\mathbb{C})$. This open subset is called the
Azumaya locus of A and denoted by ${\mathbf{azu}}_{n}A$.

One can also define the _generic Azumaya locus_ as being the
Zariski open subset of $M_n(\mathbb{C})\oplus \dots \oplus M_n(\mathbb{C})$ consisting of those tuples which generate
$M_n(\mathbb{C})$ and call this subset ${\mathbf{Azu}}_n$. In fact, one
can show that ${\mathbf{Azu}}_n$ is the Azumaya locus of a particular
order namely the trace ring of m generic $n\times n$ matrices.

What Nikolaus and Zinovy prove is that for an order A the Azumaya
locus ${\mathbf{azu}}_{n}A$ is an irreducible subvariety of
${\mathbf{Azu}}_n$ and that the embedding

${\mathbf{azu}}_{n}A\subset {\mathbf{Azu}}_n$

determines A itself! If you have
worked a bit with orders this result is strange at first until you
recognize it as being essentially a consequence of Bill Schelter's
catenarity result for affine p.i.-algebras.

On the positive
side it shows that the study of orders is roughly equivalent to that of
the study of irreducible $G{L}_n$-stable subvarieties of ${\mathbf{Azu}}_n$.
On the negative side, it shows that the $G{L}_n$-structure of
${\mathbf{Azu}}_n$ is horribly complicated. For example, it is still
unknown in general whether the quotient-variety (which is here also the
orbit space) ${\mathbf{Azu}}_n/G{L}_n$ is a rational variety.