As someone who biked daily for transportation and recreation for years, I can tell you it doesn't take any miracle to do it in street clothes.
David Richerby already mentioned that most practical bike trips are short. Mine were usually 10-20 minutes. But many times I've gone on recreational rides of 2-4 hours. And I've never owned any special "biking clothes". The most I ever thought about clothing was when my jeans started getting worn out from occasional encounters with the gears!
It probably depends on the person, but many feel perfectly fine in street clothes.
Edit: ratchet freak's answer reminded me that my bike at the time did have a big, soft seat. I don't know how much of a factor that was in feeling my clothing, but I'm sure it made those rides a lot comfier than without it.