Annual data VS Monthly data VS Quarterly data for a VAR model
For example, the response of $y_t$ to a shock in $x_t$ should be expected to be different at monthly, quarterly and annual frequencies, right?

Generally (and usually), yes.

Or it is that the results should be similar irrespective of the frequency of the data?

Generally (and usually), no. The answer to your question depends on the underlying data generating process (DGP). You can take an example process and analyze it at different levels of time aggregation analytically to figure out implications for what happens when we look at it using different frequencies. E.g. take a monthly AR(1) process
$$
y_t=\varphi_1 y_{t-1}+\varepsilon_t
$$
and see how it behaves when aggregated to quarterly or annually:
$$
\sum_{i=1}^{3}y_{t-i+1}=f\left(\sum_{i=1}^{3}y_{t-3-i+1}\right)+u_t
$$
and
$$
\sum_{i=1}^{12}y_{t-i+1}=g\left(\sum_{i=1}^{12}y_{t-12-i+1}\right)+v_t
$$
where you need to figure out $f(\cdot)$ and $g(\cdot)$ based on the original AR(1) model.