Observation Matrix in Kalman Filter
If you can do it that way, it makes it straight forward to implement the Kalman filter. As an example - consider a constant velocity model in Cartesian coordinates, but the only thing you are measuring is the position (not velocity) and you are measuring the position in a Cartesian coordinate system. Then your $\mathbf{H}$ matrix will only pick off the position value as you've described.

Now consider a Sonar or Radar system - which measure range and bearing rather than x, y, z. Now the measurements take place in a Polar/Spherical coordinate system but it is easier to keep the model for your targets as having constant velocities in the cartesian coordinate system. Now your state propagation matrix is the same as in the previous case, but now your $\mathbf{H}$ matrix needs to convert your Cartesion state coordinates to Polar/Speherical coordinates for your measurements. e,g, $R =\sqrt{x^2+y^2+z^2}$. So now your $\mathbf{H}$ matrix is not simply extracting one of the state variables. In fact your $\mathbf{H}$ is not even a constant value - it varies with the values of the state vector. Thus the transformation in the measurement matrix $\mathbf{H}$ is non-linear, so the Kalman filter is not even optimal anymore. In this case you now get into the Extended Kalman Filter to try and handle the non-linearity in the transformation.

This was meant to give you one specific example. Also, note that there are other ways of handling the Cartesian to Spherical conversion issue.