BUG: category index levels casted to non-category dtype in merge

[hliatrussellinvestments]

• I have checked that this issue has not already been reported.

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

• (optional) I have confirmed this bug exists on the master branch of pandas.

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Note: Please read this guide detailing how to provide the necessary information for us to reproduce your bug.

Code Sample, a copy-pastable example

# Your code here
import pandas as pd

df1 = pd.DataFrame({'x': [i for i in range(100)], 'y': [i for i in range(100)],
                    'z': [i for i in range(100)], 'd': [i for i in range(100)]})

df2 = df1.astype({"x":'category', "y":'category', "z":'category'})

df3 = df2.iloc[:10, :].groupby(['z', 'x'], observed=True).agg({'d': 'sum'})
df4 = df2.iloc[90:, :].groupby(['z', 'x', 'y'], observed=True).agg({'d': 'sum'})

pd.merge(df3, df4, left_index=True, right_index=True,
         how='outer').index.to_frame().dtypes

Problem description

I have 2 data frames, df3, and df4. Index of df3 is z, x and index of df4 is z, x, y.
All x, y, z are categorical, and index x is of the same category-dtype in df3 as in df4, since df3 and df4 are created from groupby on the same data frame df2. For the same reason, index z is of the same category-dtype in df3 as in df4.
So, I think after the merge, index x and z should stay as category.
But they are not, this is what I get:

z int64
x int64
y category
dtype: object

Expected Output

z category
x category
y category
dtype: object

Output of pd.show_versions()

INSTALLED VERSIONS

commit : f2ca0a2
python : 3.8.5.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.18362
machine : AMD64
processor : Intel64 Family 6 Model 94 Stepping 3, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252
pandas : 1.1.1
numpy : 1.19.0
pytz : 2020.1
dateutil : 2.8.1
pip : 20.2.3
setuptools : 47.1.0
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : 4.5.1
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 2.11.2
IPython : 7.18.1
pandas_datareader: None
bs4 : None
bottleneck : None
fsspec : None
fastparquet : None
gcsfs : None
matplotlib : 3.2.2
numexpr : None
odfpy : None
openpyxl : 3.0.3
pandas_gbq : None
pyarrow : None
pytables : None
pyxlsb : None
s3fs : None
scipy : 1.5.0
sqlalchemy : 1.3.18
tables : None
tabulate : None
xarray : None
xlrd : None
xlwt : None
numba : None

[arw2019]

This seems like a duplicate of #25412

Would you be interested in submitting a PR? Reading through #25412 there is interest in getting this to work but no one has got to it yet

[jbrockmendel]

I'm seeing category-dtype results on main. Could use a test (or determine if one already exists)

[agodinez01]

take