As expected the problem is not from the hardware but from the library!
I managed to have that shield working with Galileo Gen 2 using and modifying the Galileo library for this module :

https://www.cooking-hacks.com/documentation/tutorials/can-bus-module-shield-tutorial-for-arduino-raspberry-pi-intel-galileo/

The Galileo library is given in the middle of the page or at this address:

https://www.cooking-hacks.com/media/cooking/images/documentation/CANBus/CANBUS_for_Intel_Galileo.zip

The same library seems to work on Arduino. I have an Arduino Mega and a Galileo GEN communicating together for 15 minutes now without any problems. They both use this library and have the same program.

Hope this can help anybody who encounters the same problem!

Sources: https://www.seeed.cc/topic_detail.html?id=6135