Are the details of flight control laws the same for every fly-by-wire Airbus aircraft or are they only the same within the specific family?
From a pilot point of view they are the same, for all families.

Technically, they are differently implemented, from a family to another one, but they are implemented the same way in the same family.

The basic principle for all Airbus FBW are as follows in normal law:


in pitch normal law the side stick send a «load factor» order that would become limited when protections are triggered such as AOA protection.
in roll the side stick gives a «roll rate» that would become limited when protections are triggered such as bank angle exceeding a threshold.
When sensors redundancy is degraded the laws become degraded and we enter alternate laws or direct law.
On ground we are in direct simulated relation ship between the side stick and the related surfaces.


The differences:
They don’t affect the flying technique from the pilot point of view, they concern the protections mostly those that concern the structure of the aircraft for exemple 
The Maneuver Load Alleviation function that distributes the loads on the huge wing of the 330, this protection is specific to the 330/340