Carrying Your Papers
You must always carry some form of officially recognised identifiecation on your person. For a foreigner in Germany, this will normally be your passport. That is certainly the document that a policeman will expect to see if he stops you.
Remember though that, if you need a visa or residence permission for Germany then you should also be able to show that. Of course, a British citizen does not need a visa for Germany and so your passport is fine.
If you are living in German and you have an Aufenhaltserlaubnis that should also be sufficient.
This may seem a little harsh to somebody used to the British was in these matters but, the Germans would argue that they have little alternative.
Germany is almost completely surrounded by land borders which are very hard to keep secure. The UK is able to control immigration largely by means of border checks but this is just not possible in Germany. So, the solution is one of registration and carrying your papers.
Of course, if you look as though you have money in your pocket and are where you are for a legitimate purpose, you are unlikely to be asked for any ID but, if you do come to the attention of the police, they will expect to see it and they will be suspicious if you do not have any.