BUG: `pd.concat(..., axis="columns")` inconsistently keeps/drops index name
- [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({"IndexColumn": [1, 2, 3], "DataColumn1": ["x", "y", "z"]})
df2 = pd.DataFrame({"IndexColumn": [4, 1, 3], "DataColumn2": ["j", "k", "l"]})

df3 = pd.DataFrame({"IndexColumn": ["a", "b", "c"], "DataColumn1": ["x", "y", "z"]})
df4 = pd.DataFrame({"IndexColumn": ["d", "a", "c"], "DataColumn2": ["j", "k", "l"]})

df1_2_merged = pd.concat([df1.set_index("IndexColumn"), df2.set_index("IndexColumn")], axis="columns")
print("Integer index retained name:", df1_2_merged.index.name == "IndexColumn")
# True

df3_4_merged = pd.concat([df3.set_index("IndexColumn"), df4.set_index("IndexColumn")], axis="columns")
print("String index retains name:", df3_4_merged.index.name == "IndexColumn")
# False
```

#### Problem description

`pd.concat([df, other_df], axis="columns")` with two dataframes with identical index names sometimes retains the index name, but at other times drops it. The index values themselves are kept, and the behaviour is otherwise as expected.

It seems logical to always keep the index name if it is the same between both, rather than having inconsistent behaviour.

This _may_ be the same as #35847. However, that discusses `Index.union()`, and taking the union of the `df3` and `df4` indices directly results in the name being propagated to the output, unlike the example above:

```
>>> df3_index = df3.set_index("IndexColumn").index
>>> df4_index = df4.set_index("IndexColumn").index
>>> df3_index.union(df4_index)
Index(['a', 'b', 'c', 'd'], dtype='object', name='IndexColumn')
```

(I've not been able to test the master branch as it appears to now require Python 3.7, and I haven't got a Py3.7 environment handy currently)

#### Expected Output

In the code example above, I'd expect both to output "True", i.e. that the index name is copied through to the output.

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

<details>

INSTALLED VERSIONS
------------------
commit           : db08276bc116c438d3fdee492026f8223584c477
python           : 3.6.8.final.0
python-bits      : 64
OS               : Linux
OS-release       : 3.10.0-1127.19.1.el7.x86_64
Version          : #1 SMP Thu Aug 20 14:39:03 CDT 2020
machine          : x86_64
processor        : x86_64
byteorder        : little
LC_ALL           : None
LANG             : en_GB.UTF-8
LOCALE           : en_GB.UTF-8

pandas           : 1.1.3
numpy            : 1.18.1
pytz             : 2020.1
dateutil         : 2.8.1
pip              : 20.2.4
setuptools       : 45.2.0
Cython           : 0.29.21
pytest           : 5.4.3
hypothesis       : 5.16.0
sphinx           : 2.4.0
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : 4.5.2
html5lib         : None
pymysql          : None
psycopg2         : 2.8.4 (dt dec pq3 ext)
jinja2           : 2.11.2
IPython          : None
pandas_datareader: None
bs4              : 4.9.3
bottleneck       : 1.3.2
fsspec           : 0.6.2
fastparquet      : None
gcsfs            : None
matplotlib       : 3.1.2
numexpr          : 2.7.1
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : 0.15.1
pytables         : None
pyxlsb           : None
s3fs             : None
scipy            : 1.2.1
sqlalchemy       : None
tables           : 3.5.2
tabulate         : 0.8.6
xarray           : None
xlrd             : 1.2.0
xlwt             : None
numba            : 0.51.0

</details>
