Is this common in the states? In Denmark, employers usually ask you what you expect to be paid for the job you're interviewing for, but they never ask what you are currently being paid. Ofcourse, you should always dodge the question to let them give a number first and take it from there, but still, I've never had anyone ask me specifically what I make at my current job.
If they don't now, they will in a few years. We had the same evolution here in Belgium & the Netherlands, late 1990's or early 2000's recruiters wouldn't ask overly let alone require it. Nowadays they try hard to get you to disclose. I think one of the reasons is what is cited above, for the statistics for internal use. In time they will get more sophisticated and they'll start asking.
Aside, isn't there a website in Denmark where you can look up people's salaries? Or is that in Sweden?
As far as I know, your tax records are private. There are however surveys and a lot of unions ask members to disclose salary (with no personal information) so they give out "average wages" for specific areas, such as system development, web development, general programming, etc. (these samples were taken from one of the IT-focused unions). So it gives you an estimate of where your average should be and you can work from there.
I still haven't been aggressively asked by employers what I make now, but I have been asked aggressively what I wish for (as in, you tell us first and we'll say yes/no/maybe).
How are you to be sure what above-average salary is? You may think you are asking for above-average but instead just asking for average. Your better selling your self as above-average and use that to bump up the salary after their initial offer