To everyone reporting having their ID checked, was it train conductors checking, or the local police?
In my experience passports/ID are regularly checked over the Germany-Denmark, Spain-Portugal and Switzerland-Italy rail borders (including customs checks).
Recent events, notably in Paris and Brussels, means that extra checks are likely.
Not in Schengen (yet), but last month passports were scanned/collected on the train by police over the Hungary-Romania and Romania-Bulgaria rail borders (both sides).
As Romania and Bulgaria are not in Schengen, it would be surprising if passports were not checked when crossing those borders, just like you need to go through immigration control when crossing the English Channel.
I have twice today been asked to show ID by CD staff, travelling on an OeBB online ticket which I thought required the Credit Card as ID. However in an area where walkup tickets are the norm that may not be relevant...
I think this is more for revenue protection / policy of CD that ID is checked when using a named ticket.
Recently, I have used SBB, DB and NS named tickets and conductors did not ask for ID despite the ticket requiring ID checks, but CD always asked.
Any police officer in Belgium can stop you and ask you to prove your ID. Belgians have to carry their identity card with them at all times. Foreigners have to have a legal document with them too. It will normally not be possible to check-in in an hotel without it. Don't mess with that.
I have only been asked for ID to check into a Belgian hotel once (out of about 10 times staying overnight there). I was using a large amount of hotel reward points to get a free stay that would have been quite expensive otherwise. The front desk staff apologised for asking for ID but said that they needed to make sure that I had not stolen someone else's points, because the hotel chain had recently suffered many fraudulent free stay redemptions. My wife, who was staying with me, did not show her ID to anybody in Belgium at all.