For comparison, in the Netherlands all postcode data is open data, including detailed building outlines as well as almost all other related information.
This also leads to some very interesting issues, as third parties who automatically ingest the data have a habit of just reading the docs and making the wrong assumptions about what it means in reality.
One example I often encounter myself is Google Maps trying to geolocate my address (city, street name, house number), and then reverse-geolocate that into my postcode. Which sounds like it would work - until you realize that the postcode polygons can overlap. I live in a building where (roughly) each floor has its own postcode, so whenever I try to fill in my address on a website which uses Google's API, it'll "helpfully" auto-fill or "correct" my postcode from 1234AB to 1234AZ. It'll essentially pick a random postcode, because all of them share the same coordinates!
That's Really Really Bad, because the postcode plus house number combination is supposed to uniquely identify a mailbox: it's only a matter of luck that the house numbers aren't reused in the set of postcodes used for my building. They could've just as well reused the numbers at the individual building entrances...
This creates a very special Dutch thing —- my neighborhood had the roads on the map before the map itself was updated to show landmass instead of the body of water.
I wonder if all the houses on disconnected long islands without roads in Vinkeveense Plassen have postal codes? It's hard to get a pizza delivered there.
In the PDOK viewer linked above you can enable the "Adressen" layer[1] and it will show markers on everything that has an address. Everything that has an address has a postal code, which is listed in the details if you click the address. (There might be an exception with an address but no postal code somewhere, I'm not sure, but not here.)
That area looks so weird on a map and so cool in person. I never kinda understood what is going on there except the whole having a lake and being the Netherlands.
See https://app.pdok.nl/viewer for most datasets.