  It is known that, for the algebra of functions on a Kleinian singularity, the
parameter space of deformations and the parameter space of quantizations
coincide. We prove that, for a Kleinian singularity of type $\mathbf{A}$ or
$\mathbf{D}$, isomorphisms between the quantizations are essentially the same
as Poisson isomorphisms between deformations. In particular, the group of
automorphisms of the deformation and the quantization corresponding to the same
deformation parameter are isomorphic. We additionally describe the groups of
automorphisms as abstract groups: for type $\mathbf{A}$ they have an
amalgamated free product structure, for type $\mathbf{D}$ they are subgroups of
the groups of Dynkin diagram automorphisms. For type $\mathbf{D}$ we also
compute all the possible affine isomorphisms between deformations; this was not
known before.
