Why does metal make a sound when hammered and non-metals not?

Everything makes a sound when hit.

Rigid things tend to make more sound because the vibration propagates over the whole shape quickly because the rigid thing is held tightly, so instead of just giving way or changing shape or breaking, the thing vibrates, making a longer-lasting sound at one frequency.

Metal is a rigid thing because of its structure. See above.

A fancy wine glass will ring when struck nicely.
The hardwood blocks of marimbas make lovely tones.
Like @Zaku says a lot of material that makes sounds is rigid.

Metals are able to vibrate readily without falling apart so they will resonate at a frequency dependant on their size and shape when struck. Other things do this too but there is often a higher dampening factor.

In this discussion of top-end loudspeaker manufacture, the fellow mentions the qualities of the plastic they use and why it’s rigidity is superior to common speakers, around 13:00:
Harbin UK