Why does a metal block make a shrill sound but not a wooden block upon hammering?

Is it that the wooden block vibrates with lesser frequency than the metal block? If so, why is that?

'Yes', to the first question.

Metal is stiffer than wood and produces higher frequencies (higher pitch).

This follows from the wave equation (here in one dimension):

$$u_{tt}=\frac{E}{\rho}u_{xx}$$

$E$ is Young's Modulus and $\rho$ the material's density.

When solved, the solution contains a time-dependent factor like this:

$$\cos\Big(\frac{n\pi ct}{L}\Big)$$

where $n=1,2,3,...$, and $c=\sqrt{\frac{E}{\rho}}$ and $L$ a chracteristic length (e.g. the length of a clamped string).

To find the frequencies:

$$\cos \omega t=\cos\Big(\frac{n\pi ct}{L}\Big)$$

$$\omega=2\pi f=\frac{n\pi c}{L}$$

$$f=\frac{n}{2L}\sqrt{\frac{E}{\rho}}$$

The fundamental frequency (for $n=1$) is given by:

$$f_1=\frac{1}{2L}\sqrt{\frac{E}{\rho}}$$

So for stiffer materials, i.e. larger $E$, the fundamental frequency (as well as the harmonics) is higher.

The metal block has a relatively low level of internal damping, however the wooden block has a high level of internal damping: Much of the energy imparted to the wooden block is dissipated internally as heat and deformation, also the higher frequencies are damped more than the lower frequencies (it acts like a low pass filter).

So the wooden block will vibrate with fewer harmonics (hence less shrill) and also with lower amplitude (and also with lower duration).