Circular hyperboloid of one sheet
The surface type is not unique up to isometry or even up to similarity transformations, but rather, depends on multiple nonzero parameters . If we're considering the surface up to rigid isometries, the parameters are unique. If we're considering the surface up to similarity transformations, the parameters are unique up to projective equivalence.
Under affine transformations, a circular hyperboloid of one sheet need not remain a circular hyperboloid of one sheet. However, it is true that any two circular hyperboloids of one sheet are equivalent under affine transformations. So in this sense, the surface is unique up to affine transformations.