The OP and the parent comment are about the utility functions given by the von Neumann-Morgenstern theorem; these are left invariant by any affine rescaling and (by the uniqueness part of the theorem) are changed by any non-affine rescaling.

It's worth mentioning that all three kinds of utility functions can be constructed: ordinal scale, interval scale, and ratio scale. For an overview of ratio scale utility functions, see Peterson (2009), pp. 106-110.