Controller Area Network - RLC filter for filtering noise
It's a common mode inductor or choke therefore, the wanted signal (the CAN differential digital signal) is not affected by it. However, for common-mode interference, the filter does its job and provides attenuation or blocking.

To reiterate, with a proper CAN bus signal, equal signal currents in each coil are in opposition hence, there is no net magnetic field produced hence, it is transparent to a pure differential AC signal. No magnetic field means no inductance and no impediment to differential current flow. 

Whereas, for a common mode interference, currents are "in-phase" and the magnetic field created opposes those current changes as would a single inductor on a single line.


  First question, does the CAN protocol need a filter


No, protocols are concepts and don't need filters unless some form of modulation and demodulation is inherent in the process (which there isn't for CAN). Real world scenarios often need filters.


  So what is an harsh environnement


Vehicles are usually regarded as harsh environments for electronic signalling and that is what CAN bus was developed for.


  I would like to have your opinion about this filter. Is it well
  designed ? Under which conditions ?


If I showed you a picture of a cup: -




Could you say that it was well designed? 
Is it fit for purpose? 
Does it do what the designer intended?


If I then told you that it was designed as a chamber pot would you agree that it isn't well designed as a cup for drinking?

So, without knowing what was in the designer's mind when this filter was produced, it's impossible to say whether it was fit for purpose.


  Do you think that those oscillations can introduce error when decoding
  the signal coming from the bus ? In other words, wouldn't be better if
  the filter was not here ?


Without knowledge of the designer's intent to counter whatever common-mode interference was perceived to be problematic this cannot be answered. Yes, the performance can be deconstructed by simulation but it doesn't mean that we ever get to understand what was in the designer's mind because, the final circuit might also be a compromise solution that just about did the intended job AND that doesn't really tell us much about the real intents that drove a particular design.


  By the way, what is the purpose of using a common mode inductance
  rather than two separates inductances ?


This should be clear now; two separate inductors will not form a common mode inductor and will therefore distort the wanted differential signal.