Can a non-EU citizen travel freely within the Schengen area without passport?

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.

To travel within Schengen, you are technically still required to have travel documents on you. In addition to that, all the Scandinavian countries have had “emergency” border checks with the rest of Schengen for a few years now. You will almost certainly be asked to show your passport or ID card at the Denmark/German border, and turned back if you do not have appropriate documents.

You definitely need your passport to cross borders within the schengen area, those rules are set by each country, you may not get checked at each border but that’s a different issue in itself. Even as a citizen in a country that participated in the Schengen Agreement passports are normally carried for international travel although it might not be strictly necessary for border crossings most airlines etc will require it anyway, so your mileage may vary depending on where you dross even between the same countries and depending on how you travel

Schengen does not remove the requirement to have appropriate documentation when crossing borders (or even within a country). It only removes systematic checks at borders.

• You can still have spot checks at border points.

• There could also be “emergency measures” checks restored at some borders.

• There could even be spot checks inside a country, completely unrelated to crossing borders. The legislation for this (when/why they can do such checks, what documents you are required to have, what procedures/flexibility there might be if you don’t have them) is quite variable from country to country.

Note that regulations may require you to provide ID when checking into hotels as well, though, again, details vary a lot from country to country in terms of what documents are acceptable, what happens if you can’t provide such a document, and even whether that is enforced at all.

Exact requirements for crossing intra-schengen borders are set in national law, so it will depend on exactly where you are travelling, but in general, you will usually be required to carry a recognized travel document when crossing intra-schengen borders. For most practical purposes, this also applies to EU/EEA citizens.

In your particular case, you will be required to carry your passport both when leaving and when reentering Germany and the Scandinavian countries have similar requirements. This does not mean that you will be checked, there are no permanent checkpoints on the border, but you will end up with more or less hassle if you happen to run into a spot check. Especially on the land borders to Sweden, chances are very much real, that there will be a check. Depending on how you intend to travel, the transport operator may also refuse to transport you if you can’t show the required travel documents when checking in.

I am pretty sure that this is a duplicate and that even I recently have answered a very similar question, but I can’t find it right now.