When you change your address you can click a checkbox saying you want a protected name/address. This means that companies that have you as customers/clients won't be able to get the new address, mail won't be redirected, etc

I did it accidentally during my last move and it was a pain in the behind

And it expires after a year by default so it feels rather pointless

If you own a danish domain name (.dk), unless you have protected address, all your info is a whois lookup away. Example lookup of ft.dk (danish folketing).

  Domain:               ft.dk
  DNS:                  ft.dk
  Registered:           1995-07-04
  Expires:              2027-09-30
  Registration period:  1 year
  VID:                  no
  DNSSEC:               Signed delegation
  Status:               Active
  
  Registrant
  Handle:               DATA REDACTED
  Name:                 Folketinget
  Address:              Christiansborg Slot 1
  Postalcode:           1218
  City:                 København K
  Country:              DK
  Phone:                +4533375500
  Email:                webmaster@ft.dk
  ID status:            ID verified via electronic ID
  
  Nameservers
  Hostname:             maleah.ns.cloudflare.com
  Hostname:             yichun.ns.cloudflare.com

EDIT: I didn't redact anything from the lookup, this is the exact data I received.