I can confirm this isn't true. Danish/US dual-national right here. Denmark is fully aware of my US citizenship.

How did you become a dual national?

Born in Denmark to a Danish father and American mother. When you're born on foreign soil to an American parent that parent declares your birth to the United States and you receive a Consular Report of Birth Abroad: your US birth certificate.

I know for a fact the Danish state allows for dual-nationals because I just renewed my Danish passport and because of new biometrics requires dual-nationals to give a copy of this Report.

There's one big downside to all this, since Denmark now knows very clearly I am two people. Should the state decide to revoke this law I will most likely be forced to choose. Which would be unfortunate, since I can't imagine living in just one or the other country.