When dealing with proteins, freezing and thawing can destabilize the structure of proteins and result in loss of quality.
With lipids (fats/oils), it is a different story. Many lipids can remain in the same form irrespective of the temperature. All that happens is that the movement of the lipid's molecules slow or stop depending on how low the temperature is.
This is not a change in the structure of them but simply a slowing down of the movement of the chains of molecules. This slowing down makes oils appear stable like a solid.