A geometry in the sense of Thurston is a pair $(X,G)$ of a connected simply connected manifold and a Lie group acting transitively with compact stabilizers on $X$ (plus some other technical conditions), and a manifold locally modelled on a geometry is called ($(X,G)$-)geometric. Two geometries are said to be equivalent if there is a diffeomorphism $\phi$ between the manifolds, and a $\phi$-equivariant isomorphism of the Lie groups.
This definition is given in order to make sense of our geometric intuition of when two spaces "looks the same", and is quite weaker than the classic concept of isometric spaces. For example if we rescale a sphere $S$ by a constant $\lambda$ we get another non-isometric (affine) manifold $S_\lambda$, and the two have indeed the same geometry. In fact taken together with their isometry groups they are equivalent geometries.
Though, there appear to be other pairs of geometry representatives which are non-isometric, and even non-affine.
Question: Can anyone provide an example of two such representatives, i.e. a pair $(X_i,\operatorname{Isom}(X_i))$ of non-affine Riemannian manifolds which are geometrically equivalent?
I suspect there are no such example whitin the realm of $2$-manifolds, and I'm looking for some examples in the $Nil$ geometry.