This link does a good job explaining and giving examples of anisotropic material.
There are cases when the structure has anisotropy. But the direction of incidence is immune to it.
Slightly tangential, but taking a look into structured color in nature is really cool! For example, the morpho butterfly is a brilliant blue but has no blue pigment. What we see is from the phenomena of structured color, which is due to the nanostructures. Professor Hanrahan mentioned how in anisotropic materials, there are often these little grooves, and if you take a look into the butterfly wing structure, it has exactly this quality that allow for the blue to be seen at broad angles and for such beautiful iridescence. link
The isotrpic property is to describe BRDFs that represent reflectance properties that are invariant with respect to rotation of surface around the surface normal vector. Materials with this characteristic such as smooth plastics have isotropic BRDFs. Anisotropy, on the other hand, refers to BRDFs that describe reflectance properties that do exhibit change with respect to rotation of the surface around the surface normal vector. Some examples of materials that have anisotropic BRDFs are brushed metal, satin, and hair.