BUG: centralized helper for output coerce and na_object in stringdtype operations - #31884

PR summary

Related to #31825 but issuing it as a separate PR because the two changes don't overlap at all.

We currently have custom logic for determining the correct na_object and coerce value when operations mix StringDType instances. In some cases this can even lead to crashes. The following script segfaults for me because the na_object in the fill character array is not being handled correctly.

Python 3.14.5 (main, May 22 2026, 08:10:06) [Clang 21.0.0 (clang-2100.0.123.102)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> import numpy as np
>>> from numpy.dtypes import StringDType
>>> print(np.__version__)
2.5.1
>>> arr = np.array(["a"], dtype="T")
>>> fill = np.array("*", dtype=StringDType(na_object="*"))
>>> print(np.strings.center(arr, 5, fill))
[1]    87011 segmentation fault  python

This centralizes the logic for handling the na_object and coerce argument in ufunc loops and in the comparison machinery into one set of shared helpers.

Also adds tests for this behavior, including the case that leads to a crash above.

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AI Disclosure

I found this issue using an AI model. I also used an AI model to help with the tests and the approach for the fix.