Since GDPR has come into effect in the EU, all information on ownership and contact details for a domain are no longer publicly visible for the vast majority of domain types (TLDs). It is therefore no longer necessary to activate "Anonymous WHOIS" in order to protect your personal information.
However, in order to hide this information for .dk domains specifically, you must contact DK Hostmaster. You can read more about that here: https://www.dk-hostmaster.dk/en/anonymity