The requirement comes from Hypothesis, see
#28569 (comment)

It is necessary there to add a note to an exception describing which test case it comes from. The note should be printed by __str__ of this exception.

class Explanation(Exception):
    __module__ = "builtins"
    def __str__(self) -> str:
        return f"\n{self.args[0]}"

try:
why = "Failed!"
raise AssertionError(why)
except Exception as e:
msg = " You can reproduce this error by ...\n ..."
raise Explanation(msg) from e

    # Ideally something more like:
    e.__note__ = msg
    raise

This code shows my current best workaround based on a wrapper exception, with the traceback below annotating the additional details that I'd prefer to omit for clarity:

$ python example.py
Traceback (most recent call last):
  File "example.py", line 8, in <module>
    raise AssertionError(why)
AssertionError: Failed!
                                                                        # These lines are
The above exception was the direct cause of the following exception:    # confusing for new 
                                                                        # users, and they
Traceback (most recent call last):                                      # only exist due 
  File "example.py", line 10, in <module>                               # to implementation
    raise Explanation(msg) from e                                       # via the Explanation
Explanation:                                                            # wrapper type :-(
    You can reproduce this error by ...
    ...

The motivation for this is that we'd like to use ExceptionGroup to indicate that MultipleFailures is a group of exceptions, and replace our current print()-based method of reporting the details of the inner exceptions.

bpo-28953 is another use case for this feature.

New changeset 5bb7ef2 by Irit Katriel in branch 'main':
bpo-45607: Make it possible to enrich exception displays via setting their __note__ field (GH-29880)
5bb7ef2

Reopening to implement PEP 678.

The implementation is complete, I am leaving this open for one more documentation PR (but that should not block the release).