The primary reason metal is shiny is because it can achieve a very smooth surface. The reason we see objects is because they reflect light to some extent- if an object reflected 0 light, we would not see it! If a surface is rough (wood, concrete, paper, etc), the reflected light scatters into many directions. This means you can see the object, but no reflection.

If a surface is very smooth and polished (like metal) it reflects the light in parallel, essentially perfectly retaining the order it originally had- this allows images to be reflected. In fact, the term for this is “specular reflection,” which in Latin means “mirror-like.”