BUG: Merge with EA and MultiIndex
### 

- [X] I have checked that this issue has not already been reported.

- [X] I have confirmed this bug exists on the [latest version](https://pandas.pydata.org/docs/whatsnew/index.html) of pandas.

- [X] I have confirmed this bug exists on the master branch of pandas.


### Reproducible Example

```python
import pandas as pd

df1 = pd.DataFrame(data={("lvl0","lvl1-a"): ["1","2","3"], ("lvl0","lvl1-b"): ["4","5","6"]}, dtype=str)

df2 = pd.DataFrame(data={("lvl0","lvl1-a"): ["1","2","3"], ("lvl0","lvl1-c"): ["7","8","9"]}, dtype=pd.StringDtype())

result = pd.merge(left=df1, right=df2, on=[("lvl0","lvl1-a")])
```


### Issue Description

I am trying to make a merge between two data frames with a `MultiIndex` for the columns. The first data frame has an object data type (`str` in my example) and the second an extension array data type (`pd.StringDtype()` in my example). In this case, in an internal method, an `assign` method is used but unfortunately, the `name` of the column is not a string but a tuple...

### Expected Behavior

I would have expected the merge to happen as it is with two `str` dtype data frames or two `pd.StirngDtype()` dtype data frames.

```
    lvl0              
  lvl1-a lvl1-b lvl1-c
0      1      4      7
1      2      5      8
2      3      6      9
```

### Installed Versions

<details>

INSTALLED VERSIONS
------------------
commit           : 73c68257545b5f8530b7044f56647bd2db92e2ba
python           : 3.9.6.final.0
python-bits      : 64
OS               : Linux
OS-release       : 5.10.0-8-amd64
Version          : #1 SMP Debian 5.10.46-4 (2021-08-03)
machine          : x86_64
processor        : 
byteorder        : little
LC_ALL           : None
LANG             : fr_FR.UTF-8
LOCALE           : fr_FR.UTF-8
pandas           : 1.3.3
numpy            : 1.21.2
pytz             : 2021.1
dateutil         : 2.8.2
pip              : 21.2.4
setuptools       : 57.4.0
Cython           : 0.29.24
pytest           : 6.2.5
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : 3.0.1
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : 3.0.1
IPython          : None
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : None
fastparquet      : None
gcsfs            : None
matplotlib       : 3.4.3
numexpr          : None
odfpy            : None
openpyxl         : 3.0.9
pandas_gbq       : None
pyarrow          : None
pyxlsb           : None
s3fs             : None
scipy            : 1.7.1
sqlalchemy       : 1.4.25
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None

</details>
