I can't find any good information post-privatisation, but at least before 2013 the postcodes themselves were copyrighted by Royal Mail (likely Crown Copyright as with government data). There were attempts to enforce this in 2009[0]. I suspect the copyright is now owned by Royal Mail Group Ltd.
That aside, a practical issue is that Royal Mail still retains the rights to _allocate_ new postcodes for any new properties. Yet another failure of this particular privatisation.
That aside, a practical issue is that Royal Mail still retains the rights to _allocate_ new postcodes for any new properties. Yet another failure of this particular privatisation.
[0]: https://www.techdirt.com/2009/10/06/uk-royal-mail-uses-copyr...