I’d assume that this data would be pulled from some central government data source?
I’m also confused because in the search result a postcode (an incorrect postcode) is shown - OpenStreetMap but then after clicking through no post-code is shown - where is the incorrect one coming from?
Regrettably the authoritative postcode database in the UK is privately owned by Royal Mail, who only provide it under a very restrictive licence. Postcodes can be added to OSM manually, but there’s no way of systematically importing them.
Royal Mail won’t let us use the PAF, but there some sources released as open data, including the Office for National Statistics ONSUD and NSUL, which link to UPRNs.
A postcode wasn’t added for 21 Frobisher Road because it didn’t meet the import criteria, possibly because it is poorly aligned, or it has more than one UPRN. That doesn’t mean that the postcode can’t be associated with this property using data licensed under the Open Government Licence, just that this particular project won’d add it.
Many of the buildings in that part of Hornsey appear to have been traced before the OSMUK Land Registry polygons overlay was available and they’ll need to be realigned before they meet the import criteria. That’s a fairly easy job to do with the iD editor if you’d like to help.