While the similarities between Danish and English no doubt helped you, I think the fact that Denmark is a small country and so has less in the way of domestic media is equally important. You were exposed to English at an early age because it doesn't make economic sense to dub cartoons into a language with as few native speakers as Danish.
I'm personally not so sure. Because Denmark is a small country, it's a much closer-knit community than, say, the US. I grew up with a lot of Danish-created media, and paid as much attention to that as I did English-based media.
Any Danish kid born in the 80's (myself included) more likely than not has fond memories of watching "Bamses Billedbog" at 6pm on the weekends.
Also, plenty of cartoons are dubbed into Danish, so I'm not totally convinced about the economic argument. Denmark is a rich country with plenty of money to spend.
