Thank you @Rascas for pointg out where to ask.

For the record: the model of the joystick is Terios T3, but I have found that it is also branded as Gen Game S3.
I haven't found official docs of it, but digging in the forums I found how to set it in gamepad mode, working perfectly without the need of the extra adapter sold to work with a PC (basically turn it on holding HOME and A buttons).

Thank you @seanbeedelicious ! Works like a charm.