Why does it make sense to have noise of a different shape than the state vector?
Why does it make sense to model the noise as something which gets multiplied by a matrix $G$ in contrast to simply having a noise vector which has the same shape as the state vector?


The usual reason is because the model has some states that are not directly driven by the process noise. For example, suppose your states are:
$$
\mathbf{x}_k = \left [ \begin{array}{c} 
d_k\\
v_k\\
a_k
\end{array} \right]
$$
where $d_k$ is displacement, $v_k$ is velocity, and $a_k$ is acceleration all at time index $k$.

Suppose, as with a car, you can only change the acceleration component of the state (by pressing on the accelerator or the brake [decelerator]):

$$
\mathbf{x}_{k+1} = \mathbf{A} \mathbf{x}_{k} + 
\left [ \begin{array}{c} 
0\\
0\\
1
\end{array} \right] v_k
$$

In this case,
$$
G = \left [ \begin{array}{c} 
0\\
0\\
1
\end{array} \right]
$$