ENH: Add merge type validation on `pandas.merge`
- [x] closes #59422 (Replace xxxx with the GitHub issue number)
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature

# Tests on pandas.merge
```
(pandas-dev) kev@mac pandas % pytest pandas/tests/reshape/merge/test_merge.py
+ /opt/homebrew/Caskroom/miniforge/base/envs/pandas-dev/bin/ninja
[1/1] Generating write_version_file with a custom command
============================================================================================================ test session starts =============================================================================================================
platform darwin -- Python 3.10.14, pytest-8.3.2, pluggy-1.5.0
PyQt5 5.15.9 -- Qt runtime 5.15.8 -- Qt compiled 5.15.8
rootdir: /Users/kev/self/pandas
configfile: pyproject.toml
plugins: localserver-0.0.0, qt-4.4.0, cov-5.0.0, anyio-4.4.0, hypothesis-6.108.7, cython-0.3.1, xdist-3.6.1
collected 939 items

pandas/tests/reshape/merge/test_merge.py ...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

------------------------------------------------------------------------------------------ generated xml file: /Users/kev/self/pandas/test-data.xml ------------------------------------------------------------------------------------------
============================================================================================================ slowest 30 durations ============================================================================================================
0.02s call     pandas/tests/reshape/merge/test_merge.py::TestMergeDtypes::test_merge_ea_with_string[inner-str0]
0.02s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_non_unique_indexes
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_validation
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_indicator_result_integrity
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_indicator
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_non_unique_index_many_to_many
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_left_empty_right_notempty

(23 durations < 0.005s hidden.  Use -vv to show these durations.)
============================================================================================================ 939 passed in 1.37s =============================================================================================================
```

# Error Reproducible Example
```python
from datetime import datetime

data_A = list()
data_B = list()

for i in range(10):
    data_A.append({
        "id": i,
        "created_date": datetime.today(),
        "created_at": datetime.now(),
    })

    data_B.append({
        "id": i if i % 2 == 0 else 3*i,
        "created_date": datetime.today(),
        "created_at": datetime.now(),
    })

df_A = pd.DataFrame.from_dict(data_A)

df_B = pd.DataFrame.from_dict(data_B)

df_A.merge(df_B, how="full", on="id")
```
```
Traceback (most recent call last):
  File "/Users/kev/self/pandas/test.py", line 29, in <module>
    df_A.merge(df_B, how="full", on="id")
  File "/Users/kev/self/pandas/pandas/core/frame.py", line 10807, in merge
    return merge(
  File "/Users/kev/self/pandas/pandas/core/reshape/merge.py", line 355, in merge
    raise ValueError(f"'{how}' is not a valid Merge type ({merge_type})")
ValueError: 'full' is not a valid Merge type (['left', 'right', 'inner', 'outer', 'cross'])
```

Not sure if it should raise `ValueError` or `MergeError`

@mroeschke - Added tests and added asof as valid merge type, running `pytest pandas/tests/reshape/merge` passes all tests.

```
(pandas-dev) kev@mac pandas % pytest pandas/tests/reshape/merge
+ /opt/homebrew/Caskroom/miniforge/base/envs/pandas-dev/bin/ninja
[4/4] Linking target pandas/_libs/tslibs/offsets.cpython-310-darwin.so
============================================================================================================ test session starts =============================================================================================================
platform darwin -- Python 3.10.14, pytest-8.3.2, pluggy-1.5.0
PyQt5 5.15.9 -- Qt runtime 5.15.8 -- Qt compiled 5.15.8
rootdir: /Users/kev/self/pandas
configfile: pyproject.toml
plugins: localserver-0.0.0, qt-4.4.0, cov-5.0.0, anyio-4.4.0, hypothesis-6.108.7, cython-0.3.1, xdist-3.6.1
collected 1332 items

pandas/tests/reshape/merge/test_join.py ..........................................................................
pandas/tests/reshape/merge/test_merge.py .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
pandas/tests/reshape/merge/test_merge_asof.py .s...................................................................................................s.....................s.............................
pandas/tests/reshape/merge/test_merge_cross.py .................
pandas/tests/reshape/merge/test_merge_index_as_string.py ................................................................................
pandas/tests/reshape/merge/test_merge_ordered.py .....................
pandas/tests/reshape/merge/test_multi.py ..........................................

------------------------------------------------------------------------------------------ generated xml file: /Users/kev/self/pandas/test-data.xml ------------------------------------------------------------------------------------------
============================================================================================================ slowest 30 durations ============================================================================================================
0.02s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_full_outer_join
0.02s call     pandas/tests/reshape/merge/test_join.py::test_join_multiindex_not_alphabetical_categorical[categories0-values0]
0.02s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_non_unique_indexes
0.02s call     pandas/tests/reshape/merge/test_multi.py::TestMergeMulti::test_compress_group_combinations
0.02s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_right_outer_join
0.01s call     pandas/tests/reshape/merge/test_multi.py::TestMergeMulti::test_left_join_multi_index[True-True]
0.01s call     pandas/tests/reshape/merge/test_multi.py::TestMergeMulti::test_left_join_multi_index[False-True]
0.01s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_left_outer_join
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_validation
0.01s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_join_many_non_unique_index
0.01s setup    pandas/tests/reshape/merge/test_merge_index_as_string.py::test_merge_indexes_and_columns_lefton_righton[left_df2-right_df1-left_on3-right_on3-outer]
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_indicator_result_integrity
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_indicator
0.01s call     pandas/tests/reshape/merge/test_multi.py::TestMergeMulti::test_left_join_multi_index[True-False]
0.01s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_inner_join
0.01s setup    pandas/tests/reshape/merge/test_merge_index_as_string.py::test_merge_indexes_and_columns_lefton_righton[left_df2-right_df2-left_on1-right_on1-right]
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_non_unique_index_many_to_many
0.01s call     pandas/tests/reshape/merge/test_merge.py::TestMerge::test_merge_left_empty_right_notempty
0.01s call     pandas/tests/reshape/merge/test_multi.py::TestMergeMulti::test_left_join_multi_index[False-False]
0.01s call     pandas/tests/reshape/merge/test_join.py::TestJoin::test_join_multiindex

(10 durations < 0.005s hidden.  Use -vv to show these durations.)
====================================================================================================== 1329 passed, 3 skipped in 2.28s =======================================================================================================

``` Please view the CI logs as another test needs fixing https://github.com/pandas-dev/pandas/actions/runs/10337314281/job/28614031854?pr=59435 pre-commit.ci autofix Thanks @KevsterAmp 