It all depends on the refractive index (light scattering) of the molecules.
This is an interesting question, with it's basis in chemical composition, but the effect is really optical physics.
First, the observed effect.
How much light energy - in terms of both quantity and wavelengths - is reflected by a substance determines our observed color and "shine" characteristics. Only fairly smooth surfaces produce a "shine". Even uneven metal surfaces will not shine.
Second, the chemical reason.
The type of atomic bonding that takes place in metals produces a very smooth molecular surface. They may be unique in producing that smooth structure regardless of their macro-geometry. That is, a lump of pure metal will generally be shiny on the surface no matter what the actual shape the lump is in.
In contrast, while many minerals also exhibit extremely shiny surfaces - for example, quartz - they only do so in very specific planes of geometry due to their crystal structure.
Third, chemical differences.
Plastics are a wide variety of organic compositions. Most of them are amorphous - having no regular crystal structure. Because it is the crystal structure that creates the reflecting surface, most plastics are not as shiny as metals.