Since Airbnb needs its users to verify their ID by uploading a photo of their driver's license or passport, I'm in search for another service quite like Airbnb, but without this way of approving one's identity.
Can you recommend me some?
You have several options:
Use Airbnb without a verified ID. While a lot of hosts require one explicitly, there are still some that don't ask for it. I've tested it myself by registering a brand new user profile - it's still possible to contact hosts and make reservations without an ID scan (or even a photo).
Go the old-fashioned way and use a regular classifieds website, such as Craigslist. Countries outside the US usually have their own classifieds websites, so you'd have to do a bit of research for each region you're planning to visit.