BUG: Fix multiindex factorize extension dtypes

[davidjcastrejon]

• closes BUG: factorize does not preserve extension dtypes #62337
• Tests added and passed
• All code checks passed.
• Added an entry in the latest doc/source/whatsnew/v3.0.0.rst file

MultiIndex.factorize() was silently converting extension dtypes (Int64, boolean, string) to base dtypes, causing data corruption. This fix preserves extension dtypes by restoring them level-by-level after factorization.

Before:

import pandas as pd

mi = pd.MultiIndex.from_arrays([pd.array([1, 2, 3], dtype="Int64")])
codes, uniques = mi.factorize()
print(uniques.dtypes.iloc[0])  # int64 ← Lost extension dtype

x = pd.Series([1, None], dtype='Int32').to_frame(name='col')

# This is 'Int32Dtype()' as expected
print(pd.MultiIndex.from_frame(x).to_frame()['col'].dtype)

# This is float64
print(pd.MultiIndex.from_frame(x).factorize()[1].to_frame().iloc[:, 0].dtype)

After:

import pandas as pd

mi = pd.MultiIndex.from_arrays([pd.array([1, 2, 3], dtype="Int64")])
codes, uniques = mi.factorize()
print(uniques.dtypes.iloc[0])  # Int64 ← Extension dtype preserved

x = pd.Series([1, None], dtype='Int32').to_frame(name='col')

# This is 'Int32Dtype()' as expected
print(pd.MultiIndex.from_frame(x).to_frame()['col'].dtype)

# This is Int32Dtype()
print(pd.MultiIndex.from_frame(x).factorize()[1].to_frame().iloc[:, 0].dtype)

Performance Increase:
Some MultiIndex operations ~10% faster due to better type consistency.

Benchmarks:

asv continuous -f 1.1 upstream/main HEAD -b ^multiindex_object

[mroeschke]

Thanks for the PR but I think this is working around the core issue where algorithms.factorize is being called on self._values which is just a numpy array for a MultiIndex.

I think MultiIndex would need to override factorize and use a custom implementation if any level has an ExtentionDtype.