Presumably national CAAs in Europe are likewise part of the national governments of States
The UK CAA is not part of the UK government. It is a stautory corporation with statutory functions with regard to the regulation of civil aviation. The status of EASA is very similar except that it is established under EU legislation and is undertaking the tasks on behalf of 28 states rather than 1. Obviously the situation varies between EU Member States but I believe that one or two others follow the UK model.
So far as regualtion in the UK is concerned, in EASA's areas of [legal]competence, its status under the Chicago Convention differes little from that of the CAA.