Many to many unique combination
If I understand you correctly, your original


  Unique (name, json_map_enum, entity_json_map_fk)


Is made up of


  name: name of the attribute
  json_map_enum: the position of this attribute in the entity
  entity_json_map_fk: reference to entity


You should be able to enforce this in the mapping table as well.

CREATE TABLE Entity_Attribute_Map (
    ATTRIBUTE_JSON_MAP_fk int references ATTRIBUTE_JSON_MAP(id),
    entity_json_map_fk int references entity_json_map(id),
    JSON_MAP_ENUM int,
    Primary Key (ATTRIBUTE_JSON_MAP_fk, entity_json_map_fk, JSON_MAP_ENUM)
)