In the preface to George E. Martin's Transformation Geometry: An Introduction to Symmetry, he writes (emphasis mine)
Transformation geometry is a relatively recent expression of the successful venture of bringing together geometry and algebra. The name describes an approach as much as the content. Our subject is Euclidean geometry. Essential to the study of the plane or any mathematical system is an understanding of the transformations on that system that preserve designated features of the system.
...
The belief that geometries can be classified by their symmetry groups is no longer tenable. However, the correspondence for the classical geometries and their groups remains valid. Undergraduates should not be expected to grasp the idea of Klein's Erlanger program before encountering at least the projective and hyperbolic geometries. Therefore, although the basic spirit of the text is to begin to carry out Klein's program, little mention of the program is made within the text.
What does he mean in the bolded statement? From the Erlangen program Wikipedia page, what I understand is that there are geometric objects that have "the same symmetries" (I don't know how else to put it), but are nevertheless distinct.
Comments would be appreciated!
I suppose that he means that we now know that the symmetry group is not sufficient to capture the space. The idea of the Erlanger Program is that the group of symmetries determines what are the geometric figures and geometric properties within that space. If you have two subsets of the space that can be transformed into each other with a symmetry, then they are instances of the same type of geometric figure, and if a property is invariant under symmetry transformations, then that is a meaningful property for that given geometry.
The Lens example given by Jason De Vito in the comments illustrates how the symmetry group doesn't characterize the space