I got caught up in a security "scare" in Valencia after the Madrid train bombing (2004 ?). This was the ultimate example of "we must be seen to be doing something" - no matter how pointless !! There was full airline style security for all the long distance trains (to Madrid, Barcelona etc), with queues stretching across the station and down the roads outside.
But.........my local train back out to Xeraco didn't even have a ticket check................and it was the local commuter trains that were bombed in Madrid !
If terrorists were looking for a target they couldn't have asked for anything better than the queues in and around Valencia Station.

 
It's an odd one - at a guess the ground staff checked her in against the wrong name, but I can't figure out how it wasn't picked up at the gate. Normally it's super strict and any deviation from your passport name e.g. Matt vs Matthew can result in a no-fly or require a ticket re-issue. Perhaps BA's gate staff don't check business class tickets as carefully, but perhaps also because of the gender mismatch, the employee thought the passenger might be transgender and that any questions could get them into trouble? Also, once she got to Madrid, why on earth didn't she just go straight to a BA desk? That would have proven beyond doubt that she was in Madrid and indeed she's have the tags on her bags to prove it. 

As for the difference in the levels of checks between air and rail. I suppose it's down to two things, 1) security of the flight, 2) immigration obligations, 3) anti-smuggling, 4) treaty obligations

1) An air passenger with a weapon or bomb can (generally) do a lot more damage than their rail equivalent. Air travel is perceived to be a higher risk/profile target.
2) Airlines are delegated with the task of verifying passenger identity, right to enter the destination country and relaying that info to the security/immigration services of both. 
3) Airports are built to try and minimise smuggling, primarily of drugs
4) Countries are obliged to meet international conventions on passenger security etc
 
When it comes to rail, a train between Amsterdam and Berlin is practically little more than domestic anyway due to the Schengen area, Eurostar being a very different due to immigration and security agreements. I seem to recall the Spanish doing security checks on high-speed rail passengers due to terrorist fears and the Swedes/Danes re-instating border checks for a while for immigration purposes. 

You are right though that it does seem odd how the two are so different and how often high profile railway stations do not feel like they have been designed with security in mind. When you take the Eurostar in Paris there's a long mezzanine before the ticket barriers. I've always felt like in the event of an incident I'd be a complete sitting duck up there. 

 
BBC story of a journalist given the wrong boarding pass, not picked up by gate passes, and the kerfuffle that ensues. https://www.bbc.com/news/articles/cp3q5kqj80vo . Can't help contrasting it to the mode I'm in today - "I think I'll get on that train", find its time and put in in my planner.   Took the 10:00 out of Amsterdam which has a Berlin destination.  As Lisa is still in the Amsterdam relaxing for the day, I got off at Amersfoort and am now on a local train - oh, never mind to where, the point is the contrast in security.   Their has been a brief ticket check, but being asked for ID on these trips does happen but is rare as hen's teeth!