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