[...]
>> One 9-character string, sitting in a PNR field, threading across four organisations' financial systems.
(emphasis added)
It's not that identifiers identify. It's that an identifier identifies the same thing across multiple, independent, entirely distinct systems.
There are other examples: credit card numbers, government issued ID numbers.
But in general, identifiers have little currency outside the system that generated them, hence the "impressed" element to this.