BUG: Index.equals differs when comparing to object from numpy dtypes vs ea dtypes

[phofl]

Pandas version checks

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• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

idx1 = pd.Index([1, 2], dtype="Int64")
idx2 = pd.Index([1, 2], dtype="object")
idx1.equals(idx2)

This returns False while

idx1 = pd.Index([1, 2], dtype="int64")
idx2 = pd.Index([1, 2], dtype="object")
idx1.equals(idx2)

returns True.

Issue Description

This seems inconsistent, but not sure if this was an intended difference.

cc @jbrockmendel

Expected Behavior

I would expect both to return True, but not sure.

Installed Versions

INSTALLED VERSIONS

commit : 696e9bd
python : 3.8.12.final.0
python-bits : 64
OS : Linux
OS-release : 5.13.0-48-generic
Version : #54~20.04.1-Ubuntu SMP Thu Jun 2 23:37:17 UTC 2022
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.5.0.dev0+953.g696e9bd04f.dirty
numpy : 1.22.0
pytz : 2021.1
dateutil : 2.8.2
setuptools : 59.8.0
pip : 21.3.1
Cython : 0.29.30
pytest : 7.1.1
hypothesis : 6.23.1
sphinx : 4.2.0
blosc : None
feather : None
xlsxwriter : 3.0.1
lxml.etree : 4.6.3
html5lib : 1.1
pymysql : None
psycopg2 : None
jinja2 : 3.0.1
IPython : 7.28.0
pandas_datareader: None
bs4 : 4.10.0
bottleneck : 1.3.2
brotli :
fastparquet : 0.7.1
fsspec : 2021.11.0
gcsfs : 2021.05.0
matplotlib : 3.4.3
numba : 0.53.1
numexpr : 2.7.3
odfpy : None
openpyxl : 3.0.9
pandas_gbq : None
pyarrow : 5.0.0
pyreadstat : 1.1.2
pyxlsb : None
s3fs : 2021.11.0
scipy : 1.7.2
snappy :
sqlalchemy : 1.4.25
tables : 3.6.1
tabulate : 0.8.9
xarray : 0.18.0
xlrd : 2.0.1
xlwt : 1.3.0
zstandard : None
None

Process finished with exit code 0

[jbrockmendel]

There's an issue about how strict we want .equals to be, in particular w/r/t dtype checks.

For internal usage in the Index code it is more useful to return False in these cases. For users it should be consistent. Maybe a stricter _equals and less strict equals?

[phofl]

Thx, #46164, closing this one