= Solution
The <Airy stress function> determines the in-<plane stress> through its second derivatives. If two functions give the same <stress>, their difference $B$ has $B_{xx}=B_{xy}=B_{yy}=0$. On a connected domain this implies that its gradient is constant. Therefore
$$
\boxed{A\mapsto A+ax+by+c}
$$
is exactly the local affine gauge freedom. It also preserves biharmonicity. An imposed decay condition for $A$ can fix this freedom; adding an arbitrary biharmonic function would usually change the physical <stress> and is not a <gauge transformation>.
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