It's not legal, but many organisations still trust you are, who you say you are, if you provide name and the ID number. You can still call some banks in Denmark and get information on the account balance if you state name, account number and the ID number. Same with the tax authorities and some public authorities.
The health records are likely to include information that can be used to blackmail our politicians, business people etc. since just about everybody in Denmark uses the public health care system.