I don't think the birthday paradox is exactly the right model either. By itself, a matching fingerprint between two different rail sections on opposite sides of the world wouldn't present a problem, as long as the train knows its initial location and is smart enough to know it couldn't have instantly teleported. You just need a low enough probability of collisions between nearby pairs of track segments in order to not get lost.

But in any case, the statement you quotes smells like very sloppy statistics. There's a huge difference between saying that the pattern of permeability is unpredictable, and saying it's random -- the latter is a very strong assumption. It's easy to imagine ways the measurement could have local or global biases towards particular kinds of patterns, and the existence of any such effects would make the "one in a trillion" number completely meaningless.

On top of that, the article says magnetic permeability isn't part of the official material specifications. What happens if the railroad operator switches to a new supplier, whose process results in rails that are much more uniform?

This seems to be the research group's web page: https://www.mrt.kit.edu/disensor/ I looked for a published paper that would go into more detail, but I couldn't find one.

If there were biases, then the pattern of permeability would be predictable. For instance, encrypted data is not random, but it's not predictable.

The birthday paradox has a factor for the number of people in the room - the question isn’t usually who else has the same birthday in the entire universe, it’s who else within loud talking distance.

Here restricting the search to a radius of known starting positions should suffice.

These sections would be sequential making the search space very small.