Not all physical quantities are eigenvalues of some operator, time being the most immediate example.
When a quantum system is described by a wave function, the time evolution is determined by a unitary operator that acts on the Hilbert space used to describe the particle. If the Hilbert space has dimension "n", then the time evolution operator over a given time interval is an element of U(n). Now, physics enters because we do not necessarily have the capacity to reproduce the effect of an arbitrary element of U(n). A physical interaction will have the effect of causing the system to move within some continuous subgroup S of U(n). An interaction over an infinitesimal interval of time gives rise to an element of the Lie algebra "s" associated with S, which is a vector in the tangent space of the identity of S. By our assumptions, this "s" can be used to build physical Hamiltonian operators, and acts on the Hilbert space in a way that is consistent with the action of S.
The Hilbert space used to describe the wave function decomposes into eigenspaces under the action of 's', and these eigenspaces are 'distinguishable' because they generally have different dynamical properties (consider the differences between singlet and triplet states in a two-electron system: one exhibits entanglement in all reference frames, while the other does not). The actual eigenvalues of 's' can be determined in the finite-dimensional case through verifying certain global symmetries of the system, or through measuring the density of states in an experimental setup where the microcanonical ensemble Hamiltonian is given by 's'.