1) "Have your passport on your person": Yes, unfortunately this country requires you to be able to provide some ID. No, you don't need to carry it with you. If you DON'T have one though, it might cause issues/trouble/cost money. [1] I assume the Germans were misinformed and scared, the anecdote was probably not true.
2) You have to show your ID or passport at the border (aka. the airport in this case). You don't need to do that if you move in the Schengen area: Travelling from Germany to the Netherlands won't require an ID. That said, maybe you talk about braindead boarding procedures only. Those exist. People like to make sure that you're the one that checked in and all and .. yeah.. Please provide your boarding pass and ID. Useless. Enforced randomly. Utterly insane. Consider it a nuisance if it happens, but .. it doesn't, most of the time.
3) You never need to show an ID in a train. Except: a) you cross a border and leave the Schengen area. Normal procedures, people will check your passport/ID b) you bought a ticket that says 'can be used in a train (as paper or digital), but requires an ID'. That's the DB (German Railway company) being insane, stupid, annoying, crap, (insert-more-expletives-here). You can buy a ticket at a counter and no one is ever going to ask you for your ID. And I guess no one knows why they consider it a reasonable thing to do for most online tickets/digital tickets.
So:
1) Wrong 2) Maybe 3) DB is run by idiots, is crap, sucks - and there is a workaround
1: Seems to be German only, Google translate? http://de.wikipedia.org/wiki/Personalausweisgesetz