BUG: DataFrame index name is missing after call ".loc[]" in some cases.
- [X] I have checked that this issue has not already been reported.

- [X] 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.

---

#### Code Sample, a copy-pastable example

```python
import pandas as pd

df1 = pd.DataFrame({"col1": [1, 2, 3], "col2": [2, 3, 4], "col_index": ['a', 'b', 'b']}).set_index("col_index")
df2 = pd.DataFrame({"col1": [1, 2, 3], "col2": [2, 3, 4], "col_index": ['a', 'b', 'c']}).set_index("col_index")

selected_index = df1.index.intersection(['a'])
print(df1.loc[selected_index])
print(df2.loc[selected_index])

```

#### Problem description

These two dataframes are almost the same, except the last index value. However, after we select the first row with index as "a" by `.loc[]` on both dataframes, the returned dataframes are different. And the index name of the result for `df2` is missing, which is unexpected.
```
>>> print(df1.loc[selected_index])
           col1  col2
col_index
a             1     2

>>> print(df2.loc[selected_index])
   col1  col2
a     1     2
```

#### Expected Output
The index name for the `df.loc[selected_index])` should be `col_index` as it is for `df1`.


#### Output of ``pd.show_versions()``

<details>

>>> pd.show_versions()

INSTALLED VERSIONS
------------------
commit           : b5958ee1999e9aead1938c0bba2b674378807b3d
python           : 3.6.8.final.0
python-bits      : 64
OS               : Windows
OS-release       : 10
Version          : 10.0.19041
machine          : AMD64
processor        : Intel64 Family 6 Model 85 Stepping 4, GenuineIntel
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : None.None

pandas           : 1.1.5
numpy            : 1.18.5
pytz             : 2019.1
dateutil         : 2.8.1
pip              : 20.2.4
setuptools       : 52.0.0.post20210125
Cython           : 0.29.14
pytest           : 6.2.2
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : 2.11.3
IPython          : 7.16.1
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : 3.1.3
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : 1.0.1
pytables         : None
pyxlsb           : None
s3fs             : None
scipy            : 1.4.1
sqlalchemy       : None
tables           : None
tabulate         : 0.8.9
xarray           : None
xlrd             : None
xlwt             : None
numba            : None

</details>
