I haven't been to Denmark, but I did visit Munich a couple of years ago and they have a very significant biking population. The thing is, as far as I could tell the residents of Munich weren't using bicycling as a substitute for driving so much as they were using it as a substitute for walking. Running errands around town? A bike works for that. Heading to the train station? A bike works for that. Going more than a few miles where a train won't help? Take the car. I'm obviously painting with a broad brush, but that's what it looked like to me.
Now back to my situation in the states.... I live in one suburb and work in another. My home and work are 10 miles apart. I could ride my bike 2 miles to the local train station, leave my bike there, hop a train to Hillsboro and ride the company shuttle to the office. I could take my bike on the train and ride the last mile and a half, though the trains are only meant to carry about 8 bikes so that can be problematic. I could get a folding bike and carry it on the train. None of these options would require special clothing. I can easily ride 3 miles without sweating most days.
However, I find it much more convenient, not to mention considerably more enjoyable, to ride the entire 10 miles to work. To do that at the pace I enjoy riding, my every day clothing doesn't really work. So I wear bike-specific stuff.