Can you do it? Yes, probably. You will probably just get through, and nobody even notices or cares. I've not been controlled in the street in almost 5 decades, and I've not been controlled on an EU border (including non-Schengen, very much to my surprise!) in... don't know how many years. 15 years at least.
Should you do it? No. Germany in particular requires you to be able to present a valid identification document, as does crossing a Schengen border. This applies to everybody, not just foreigners.
Literally, German law says "be in possession of" (besitzen) which is technically not the same as "own one, in general". Besitzen literally means "exercise physical control over something". A thief stealing your car is the Besitzer, for example. Which is just silly.
That's probably just a stupid wording in a law made by stupid people (or a reference to the fact that the passport itself has written "Property of Country XYZ" on it when you paid for it, and it's clearly yours, with your photo inside). But whatever it is, reading the law by the letter you're breaking the law the moment you leave your passport at home (or in an embassy) and go through the street, since you no longer possess the passport.
Now, abstaining from these own vs possess sophistries, getting a bit more practical: The law requires you to carry identification when crossing the border, but it does not strictly require you to carry it at all times otherwise. It requires you to have ("own") one.
However, while you are not required to carry ID at all times, you are required to present it timely upon being challenged by police or another authorized instance (court, office, whatever). The meaning of timely is up to debate, and it can cost you 3,000 Euros if whatever time it takes isn't "timely enough" in the officer's opinion. The understanding that lawyers and such usually have of "timely" is "immediately, until something of highest importance prevents you from doing so", as in: you cannot comply now or else desastrous stuff would happen.
But whatever, if you just piss off the officer enough (or if he already had a bad enough day), the meaning may be something quite different, and you have very few options there.
So... I wouldn't want to stress my luck if that's not necessary. It means needlessly taking a risk.