Apparently no one cares, until it becomes a financial issue. IT professionels have pointed out that the system is deeply flawed for 15 - 20 years, at least, but every issue has been papered over with more IT, tweaks to software and websites. The fundamental issues have never been addressed.
The average Dane doesn't even care. They'll just complain that they need to scan their health card, rather than shouting their CPR number across the pharmacy. Thousands of people have access to the system every day, abuse happens daily, but no one seems to care, because there hasn't been an actual costs associated with that abuse.
If you start thinking more about this, more and more problems pop up.
The CPR alone is used for casual identification.
I don't think any IT infrastructure is doing it, it's all by a national single-sign on system.