I think that this is a subtlety different issue.
There is already an "ID Number" field in the user profile which Moodle anticipates it use being for integrating with external systems.
The wider problem (and we've had similar issues) is that this ID Number *isn't* displayed anywhere.
It maybe that it's better raised as a separate issue that the institution-specific user identifier is displayed alongside full names.
We have a further problem along these lines that *every* user has a person code that uniquely identifies them, but it isn't widely used, but *students* also have a Registration (matriculation) number that is used frequently.
However since we integrate using person code, we have to store the Registration Number in a custom profile field, which is even harder to have displayed every where.
(my current thought is that you *could* override the fullname() function to display this information but it's a bit hacky and breaks table sorting)