Do EEA countries differ in that regard?
Yes, they differ a lot, from making it mandatory to carry a specific form of ID at all times to not issuing ID documents to citizens at all unless they want to travel internationally.
For the sake of example, can a Dutch citizen living in Germany ask to be issued an ID card in the German format and a residence permit/PR, even if he does not need them and in case he wants them?
That's a very contrived scenario and I have never heard of anything like that. What's not uncommon is that a residence card or permit can effectively serve as an ID for all common purposes in the country where it was issued. That's useful if you come from a country where passports are expensive, have very short validity, and only serve as an income stream for the embassy.
In the Netherlands, you can vote (for EU citizens) or accomplish any official formality, in France, you can pick up a parcel, work, etc. only with the residence permit. That's usually not the case in Germany, though. Germany is particularly rigid about the need to hold a valid passport together with your residence permit at all times. A valid passport is also required to renew your residence permit and if I am not mistaken, the validity of the permit will be restricted to coincide with that of your passport.