BUG: Index.equals different behavior for EA vs non-EA types

[lukemanley]

Pandas version checks

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• I have confirmed this bug exists on the latest version of pandas.

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

Reproducible Example

import pandas as pd

f32 = pd.Index([1,2,3], dtype='float32')
f64 = pd.Index([1,2,3], dtype='float64')
F32 = pd.Index([1,2,3], dtype='Float32')
F64 = pd.Index([1,2,3], dtype='Float64')

>>> f32.equals(f64)
True

>>> F32.equals(F64)
False

Issue Description

The docs for Index.equals state that the dtype is not part of the equality test between indexes. The last example in the docs shows indexes with different dtypes being considered equal.

https://pandas.pydata.org/docs/reference/api/pandas.Index.equals.html

However, in the case of EA types the behavior is different and dtype does seem to be part of the equality check.

Expected Behavior

I'm not sure whether the different behavior between EA and non-EA types is actually a bug. Maybe it is expected in which case it should be mentioned in the docs?

Installed Versions

INSTALLED VERSIONS

commit : 90d32204ced71ea9f93fee3e271a6081d25a73c8
python : 3.8.12.final.0
python-bits : 64
OS : Linux
OS-release : 5.10.16.3-microsoft-standard-WSL2
Version : #1 SMP Fri Apr 2 22:23:49 UTC 2021
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : C.UTF-8
LOCALE : en_US.UTF-8

pandas : 1.5.0.dev0+449.g90d32204ce
numpy : 1.21.5
pytz : 2021.3
dateutil : 2.8.2
pip : 21.3.1
setuptools : 60.5.0
Cython : 0.29.26
pytest : 6.2.5
hypothesis : 6.36.0
sphinx : 4.4.0
blosc : None
feather : None
xlsxwriter : 3.0.2
lxml.etree : 4.7.1
html5lib : 1.1
pymysql : None
psycopg2 : None
jinja2 : 3.0.3
IPython : 8.0.1
pandas_datareader: None
bs4 : 4.10.0
bottleneck : 1.3.2
fastparquet : 0.7.2
fsspec : 2021.11.0
gcsfs : 2021.11.0
markupsafe : 2.0.1
matplotlib : 3.5.1
numba : 0.55.0
numexpr : 2.8.0
odfpy : None
openpyxl : 3.0.9
pandas_gbq : None
pyarrow : 6.0.1
pyreadstat : 1.1.4
pyxlsb : None
s3fs : 2021.11.0
scipy : 1.7.3
sqlalchemy : 1.4.31
tables : 3.6.1
tabulate : 0.8.9
xarray : 0.18.2
xlrd : 2.0.1
xlwt : 1.3.0
zstandard : None

[phofl]

This does look intentional (in ExtensionArray.equals) and is documented as well in the ea method.

https://pandas.pydata.org/docs/reference/api/pandas.api.extensions.ExtensionArray.equals.html

But not sure if this should leak to Index.equals

cc @jbrockmendel

[jbrockmendel]

unclear xref #33940

[phofl]

Closing in favor of #33940