Muscles sore and ridicously stiff after riding After buying my first bike I've probably not missed a single day without sitting on the bike.

I've been probably riding 10km per day after being somewhat out of shape.

Over the days I've noticed that I lose my breath faster and faster, and my muscles seem to stiff up.

This does not hurt or feel like "postworkout"-soreness, but more like passive cramping.

That is probably the best way to describe my problem. 
I'll probably see a doc anyways but I want some more input on that
Have think about your bike fit, I once had horrendous ITB pain from setting my seatpost just a few mm too high. There's plenty of conflicting advice and opinions you can research.

As well, are you doing any pre/post ride stretching or body care? I use a foam roller on my legs, just a few minutes in the morning and evening (I commute to work &amp; back) does wonders.

Finally, rest days are important! If you're going every day from a history of not riding, kick back and let your body repair itself.