EMA and OMA are two methods used for determining the mode shapes of the natural frequencies of mechanical systems. I understand it until that point. I know that for EMA we need an artificial excitation like a hammer and measure the response. But what is the final objective of these methods? Is it to model the dynamical system or to obtain the parameters? Or maybe it is a vibration analysis or measurement method? It´s not quite clear what the answer is.
Typically you are not doing modal analysis for its own sake. Typically you have a problem and you are using modal analysis as a tool to solve that problem. The goal of the modal analysis will depend on what exactly your problem is. Here are some examples
1) You have a part which is exposed to vibration, and is failing (cracking) due to high cycle fatigue. Often in these cases it is because the part has a natural frequency that is close to the excitation frequency, and so the stresses in the part are being amplified by resonance. One possible fix is to redesign the part to raise the natural frequency away from the excitation frequency. Understanding the mode shape will help you understand the best way to raise the natural frequency to avoid the problem.
2) Your customers are complaining that your device is too loud / generating too much noise. A modal analysis will help you understand how the vibration is being transmitted / radiated into sound, and thus help you redesign the part to reduce the noise.
The most basic purpose of the measurements is system identification - i.e. to determine the modal frequencies, mode shapes, and damping factors for the modes in some frequency range.
You can use that information for many different purposes. If you measure it in sufficient detail, you can construct a mathematical model of the structure that can be used in the same way as finite element model. Alternatively, you can use the measurements to verify that a FE model is accurate, or fine-tune the FE model to improve its accuracy. Creating a FE model "from first princples" may be inaccurate if the structure contains joints with unknown flexibility, etc.
Usually there is no way to model damping accurately from first principles, but measured damping ratios can be included in a finite element model to improve on "guesses" of the damping level based on general experience of how similar structures behave.
Note that with modern measurement techniques, you can sometimes measure in more detail than a finite element model will provide. For example, using a scanning laser Doppler vibrometer, you can quickly measure the vibration response of literally thousands of points on a machine such as an engine or gearbox while it is operating - though there are some limitations in what is measureable, for example the laser beam has to be able to "see" the part being measured.