if I want to book a train, let's say, from Amsterdam to Brussels or from Brussels to Cologne, can I buy a ticket without need to show my EU card and hence without being registered in a system? And if I can, are there police checks that register my identity?
Train travel is essentially anonymous. If you buy a ticket at a manned ticket counter, or from a ticket vending machine you will not be asked to show an ID, nor will you be required to show an ID when boarding a train.
The railways do not care who travels on their trains. Trains do not have passenger manifests.
They do care that everyone on their trains has paid for the privilege. So they do care that your ticket is real. With tickets printed at a ticket office, or in a ticket vending machine this is not an issue, as these tickets are printed on real ticket stock. However with print-at-home tickets there is the real risk that people would start photoshopping their own tickets, or printing multiple copies and sharing them with friends.
So for print at home tickets a name will be asked. This does not mean you have to register (although some railways encourage this), but it does mean that you need to have a piece of ID with you (as you need in most EU countries anyway). The purpose here is to avoid counterfeiting and copying. The name you give when booking is encoded in the ticket, in a way that prevents you from creating the ticket yourself. And because your name is on the ticket it is only useful for you.
But the purpose is not to keep track of your movements. It is just an anti fraud measure.
In my experience, only the German railway seems specific about the type of ID you should carry and enforcing these provisions at all. Travelling regularly with e-tickets on Thalys or French trains, I have never ever have to show ID. Never. I have also never seen a train guard ask anyone for ID (and only very rarely seen the police do it). Nov 13, 2019 at 23:22
When you are running a service with compulsory reservation ticket control is easier. On Thalys often tickets aren't even checked. if the number of people on board equals the number of tickets sold for that train then everyone on board has paid... DB sells most tickets without reservations, thus is more exposed to the risk of people copying tickets... Hence they ask for an ID (and so do for example the SBB) Nov 14, 2019 at 12:00
@KristvanBesien It does make a lot of sense... yet it's not completely true either in practice! I experience a ticket check almost every time on Thalys and French train guards have to check tickets at least once during a trip. Nov 14, 2019 at 19:46
There may be police checks at any point, but there is no routine need to prove identity on most intra-EU trains.
(Note temporary exceptions to this may include trains between Italy and France, Denmark and Sweden, and German print-at-home tickets do require ID.)
German print-at-home tickets do require ID This is also true for print at home and electronic tickets on Thalys, which is the operator for the routes the question mentioned. Nov 13, 2019 at 15:08
This does not answer the OPs question. He is inquiring about the need to show an ID when buying a ticket, not during travel itself. Nov 13, 2019 at 15:34
1@MJeffryes In theory only, I use it once a month or more and I never ever had to show ID. Also, between Amsterdam and Brussels there is also a regular NS train which is little more than a glorified suburban train (+1 to the answer) Nov 13, 2019 at 23:22