English translation: Central Population Registry number
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Danish Person Number / Danish CPR no. / Danish ID no.
Explanation: I've seen "Personnummer" written on offical documents in the place of CPR-nummer. CPR-nr. stands for: Personnummeret i det central personregister. Directly in English i.e. The Person Number in Denmarks Central Person Register. i.e. Danish CPR no.
However Person Number in English is better described as Personal Identity Number or ID no.
Social Security or Tax File Number are equivalent numbers used in English Speaking Countries (USA and Australia).
Danish ID no. is another suggestion. Drop Danish if it's obvious the number pertains to Denmark.