Postcodes in Nominatim
The blog post Nominatim and Postcodes describes the handling implemented since Nominatim 3.1.
Postcode centroids (aka 'calculated postcodes') are generated by looking at all postcodes of a country, grouping them and calculating the geometric centroid. There is currently no logic to deal with extreme outliers (typos or other mistakes in OSM data). There is also no check if a postcodes adheres to a country's format, e.g. if Swiss postcodes are 4 digits.
Regular updating calculated postcodes
The script to rerun the calculation is
nominatim refresh --postcodes
and runs once per night on nominatim.openstreetmap.org.
Finding places that share a specific postcode
In the Nominatim database run
SELECT address->'postcode' as pc, osm_type, osm_id, class, type, st_x(centroid) as lon, st_y(centroid) as lat FROM placex WHERE country_code='fr' AND upper(trim (both ' ' from address->'postcode')) = '33210';
Alternatively on Overpass run the following query
[out:json][timeout:250]; area["name"="France"]->.boundaryarea; ( nwr(area.boundaryarea)["addr:postcode"="33210"]; ); out body; >; out skel qt;