BUG: pd.Index.Intersection fails with multiple data frames

[Ge0rges]

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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

 df1 = pd.DataFrame({"name": ["1", "3", "c"], "value": [1, 2, 3]})
 df2 = pd.DataFrame({"name": ["b", "c", "d"], "value": [4, 5, 6]})
 df3 = pd.DataFrame({"name": ["c", "d", "e"], "value": [7, 8, 9]})

 df1.set_index("name", inplace=True)
 df2.set_index("name", inplace=True)
 df3.set_index("name", inplace=True)

 common_index = df1.index.intersection([df1.index, df2.index, df3.index])

### Issue Description

Intersection unexpectedly crashes with traceback:

Traceback (most recent call last):
File "/Applications/PyCharm.app/Contents/plugins/python/helpers/pydev/pydevd.py", line 1534, in _exec
pydev_imports.execfile(file, globals, locals) # execute the script
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/Applications/PyCharm.app/Contents/plugins/python/helpers/pydev/_pydev_imps/_pydev_execfile.py", line 18, in execfile
exec(compile(contents+"\n", file, 'exec'), glob, loc)
File "/Users/GeorgesKanaan/Documents/Development/Methylation/willis_dmr_analysis.py", line 158, in
common_index = df1.index.intersection([df1.index, df2.index, df3.index])
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/Users/GeorgesKanaan/micromamba/envs/jupyter/lib/python3.12/site-packages/pandas/core/indexes/base.py", line 3532, in intersection
result = self._intersection(other, sort=sort)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/Users/GeorgesKanaan/micromamba/envs/jupyter/lib/python3.12/site-packages/pandas/core/indexes/base.py", line 3561, in _intersection
res_values = self._intersection_via_get_indexer(other, sort=sort)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/Users/GeorgesKanaan/micromamba/envs/jupyter/lib/python3.12/site-packages/pandas/core/indexes/base.py", line 3582, in _intersection_via_get_indexer
right_unique = other.unique()
^^^^^^^^^^^^^^
File "/Users/GeorgesKanaan/micromamba/envs/jupyter/lib/python3.12/site-packages/pandas/core/indexes/base.py", line 3065, in unique
if self.is_unique:
^^^^^^^^^^^^^^
File "properties.pyx", line 36, in pandas._libs.properties.CachedProperty.get
File "/Users/GeorgesKanaan/micromamba/envs/jupyter/lib/python3.12/site-packages/pandas/core/indexes/base.py", line 2346, in is_unique
return self._engine.is_unique
^^^^^^^^^^^^^^^^^^^^^^
File "index.pyx", line 266, in pandas._libs.index.IndexEngine.is_unique.get
File "index.pyx", line 271, in pandas._libs.index.IndexEngine._do_unique_check
File "index.pyx", line 333, in pandas._libs.index.IndexEngine._ensure_mapping_populated
File "pandas/_libs/hashtable_class_helper.pxi", line 7115, in pandas._libs.hashtable.PyObjectHashTable.map_locations
TypeError: unhashable type: 'Index'

### Expected Behavior

I should get out a list of indices common to all the indices passed.

### Installed Versions

<details>

INSTALLED VERSIONS
------------------
commit                : d9cdd2ee5a58015ef6f4d15c7226110c9aab8140
python                : 3.12.3.final.0
python-bits           : 64
OS                    : Darwin
OS-release            : 23.5.0
Version               : Darwin Kernel Version 23.5.0: Wed May  1 20:16:51 PDT 2024; root:xnu-10063.121.3~5/RELEASE_ARM64_T8103
machine               : arm64
processor             : arm
byteorder             : little
LC_ALL                : None
LANG                  : en_US.UTF-8
LOCALE                : en_US.UTF-8

pandas                : 2.2.2
numpy                 : 1.26.4
pytz                  : 2023.3.post1
dateutil              : 2.8.2
setuptools            : 68.2.2
pip                   : 23.3.1
Cython                : 3.0.10
pytest                : None
hypothesis            : None
sphinx                : None
blosc                 : None
feather               : None
xlsxwriter            : None
lxml.etree            : None
html5lib              : None
pymysql               : None
psycopg2              : None
jinja2                : 3.1.2
IPython               : 8.17.2
pandas_datareader     : None
adbc-driver-postgresql: None
adbc-driver-sqlite    : None
bs4                   : 4.12.2
bottleneck            : None
dataframe-api-compat  : None
fastparquet           : None
fsspec                : None
gcsfs                 : None
matplotlib            : 3.8.4
numba                 : None
numexpr               : None
odfpy                 : None
openpyxl              : 3.1.2
pandas_gbq            : None
pyarrow               : None
pyreadstat            : None
python-calamine       : None
pyxlsb                : None
s3fs                  : None
scipy                 : 1.13.0
sqlalchemy            : None
tables                : None
tabulate              : None
xarray                : None
xlrd                  : None
zstandard             : None
tzdata                : 2023.3
qtpy                  : None
pyqt5                 : None
</details>

[Aloqeely]

Thanks for the report! Index.intersection only compares 2 Index objects at a time, so you can't directly call it to compare 3 or more indexes, you can however chain the intersection method (call it multiple times on the result) to find the intersection of more than two indexes.

common_index = df1.index.intersection(df2.index).intersection(df3.index)

[tiago-firmino]

take

[Ge0rges]

Thank you for the clarification I misunderstood the docs when the input mentioned array-like my mistake.