Describe the issue:

Creating a StringDType ndarray with na_object using full (and full_like) coerces the nan sentinel to a string.

I can work around this using arr[:] = np.nan, but think the behavior is unexpected.

Reproduce the code example:

import numpy as np

arr1 = np.full((1,), fill_value=np.nan, dtype=np.dtypes.StringDType(na_object=np.nan))

arr2 = np.full_like(arr1, fill_value=np.nan)

assert arr1.item() is np.nan
assert arr2.item() is np.nan

Error message:

Traceback (most recent call last):
  File "/Users/goldbaum/Documents/numpy/../numpy-experiments/test.py", line 7, in <module>
    assert arr1.item() is np.nan
           ^^^^^^^^^^^^^^^^^^^^^
AssertionError

Python and NumPy Versions:

2.2.1
3.12.8 | packaged by conda-forge | (main, Dec 5 2024, 14:24:40) [GCC 13.3.0]

Runtime Environment:

[{'numpy_version': '2.2.1',
'python': '3.12.8 | packaged by conda-forge | (main, Dec 5 2024, 14:24:40) '
'[GCC 13.3.0]',
'uname': uname_result(system='Linux', node='poisson', release='6.8.0-51-generic', version='#52~22.04.1-Ubuntu SMP PREEMPT_DYNAMIC Mon Dec 9 15:00:52 UTC 2', machine='x86_64')},
{'simd_extensions': {'baseline': ['SSE', 'SSE2', 'SSE3'],
'found': ['SSSE3',
'SSE41',
'POPCNT',
'SSE42',
'AVX',
'F16C',
'FMA3',
'AVX2'],
'not_found': ['AVX512F',
'AVX512CD',
'AVX512_KNL',
'AVX512_KNM',
'AVX512_SKX',
'AVX512_CLX',
'AVX512_CNL',
'AVX512_ICL',
'AVX512_SPR']}},
{'architecture': 'Haswell',
'filepath': '/home/mathause/.conda/envs/regionmask_dev/lib/libopenblasp-r0.3.28.so',
'internal_api': 'openblas',
'num_threads': 8,
'prefix': 'libopenblas',
'threading_layer': 'pthreads',
'user_api': 'blas',
'version': '0.3.28'}]

Context for the issue:

No response