BUG: left merge where right frame has different dtype key column has mismatching result dtypes compared to a "reversed" right merge

[wence-]

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• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd

a = pd.DataFrame({"key": pd.Series([1, 2], dtype="int64")})
b = pd.DataFrame({"key": pd.Series([1.0, 2.0], dtype="float64")})

left_merge_dtypes = a.merge(b, on="key", how="left").dtypes # int64

# Now we'll do a right merge with the arguments swapped
right_merge_dtypes = b.merge(a, on="key", how="right").dtypes # float64

print(left_merge_dtypes.equals(right_merge_dtypes))
# False

Issue Description

A "left" merge uses keys from the left frame, whereas a "right" merge uses keys from the right frame. If we also swap the order of the frame arguments to the merge, up to suffix naming, I expect that a.merge(b, how="left") <=> b.merge(a, how="right").

The result is the same, up to the dtype of the returned frames, which was surprising.

Expected Behavior

I would not expect different dtypes in these two cases. It looks like the merged key column is in this case taken from the "left" frame, ignoring how.

Installed Versions

INSTALLED VERSIONS
------------------
commit                : b864b8eb54605124827e6af84aa7fbc816cfc1e0
python                : 3.10.12.final.0
python-bits           : 64
OS                    : Linux
OS-release            : 6.2.0-37-generic
Version               : #38~22.04.1-Ubuntu SMP PREEMPT_DYNAMIC Thu Nov  2 18:01:13 UTC 2
machine               : x86_64
processor             : x86_64
byteorder             : little
LC_ALL                : None
LANG                  : en_GB.UTF-8
LOCALE                : en_GB.UTF-8

pandas                : 2.1.0.dev0+1414.gb864b8eb54
numpy                 : 1.26.2
pytz                  : 2023.3
dateutil              : 2.8.2
setuptools            : 67.7.2
pip                   : 23.2.1
Cython                : 3.0.0
pytest                : 7.4.0
hypothesis            : 6.82.2
sphinx                : 6.2.1
blosc                 : None
feather               : None
xlsxwriter            : 3.1.2
lxml.etree            : 4.9.3
html5lib              : 1.1
pymysql               : 1.4.6
psycopg2              : 2.9.9
jinja2                : 3.1.2
IPython               : 8.14.0
pandas_datareader     : None
adbc-driver-postgresql: None
adbc-driver-sqlite    : None
bs4                   : 4.12.2
bottleneck            : 1.3.7
dataframe-api-compat  : None
fastparquet           : 2023.7.0
fsspec                : 2023.6.0
gcsfs                 : 2023.6.0
matplotlib            : 3.7.2
numba                 : 0.58.1
numexpr               : 2.8.4
odfpy                 : None
openpyxl              : 3.1.2
pandas_gbq            : None
pyarrow               : 13.0.0
pyreadstat            : 1.2.2
python-calamine       : None
pyxlsb                : 1.0.10
s3fs                  : 2023.6.0
scipy                 : 1.11.1
sqlalchemy            : 2.0.19
tables                : 3.8.0
tabulate              : 0.9.0
xarray                : 2023.7.0
xlrd                  : 2.0.1
zstandard             : 0.22.0
tzdata                : 2023.3
qtpy                  : 2.4.0
pyqt5                 : None