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?
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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.