diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index b7efec8fd2e89..b3aa2ac95880d 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -545,6 +545,7 @@ Reshaping
 - Bug in :func:`crosstab` when inputs are are categorical Series, there are categories that are not present in one or both of the Series, and ``margins=True``. Previously the margin value for missing categories was ``NaN``. It is now correctly reported as 0 (:issue:`43505`)
 - Bug in :func:`concat` would fail when the ``objs`` argument all had the same index and the ``keys`` argument contained duplicates (:issue:`43595`)
 - Bug in :func:`concat` which ignored the ``sort`` parameter (:issue:`43375`)
+- Fixed bug in :func:`merge` with :class:`MultiIndex` as column index for the ``on`` argument returning an error when assigning a column internally (:issue:`43734`)
 
 Sparse
 ^^^^^^
diff --git a/pandas/core/reshape/merge.py b/pandas/core/reshape/merge.py
index fa09f003bc7b8..a88d1dce693f6 100644
--- a/pandas/core/reshape/merge.py
+++ b/pandas/core/reshape/merge.py
@@ -1280,10 +1280,12 @@ def _maybe_coerce_merge_keys(self) -> None:
             # incompatible dtypes. See GH 16900.
             if name in self.left.columns:
                 typ = lk.categories.dtype if lk_is_cat else object
-                self.left = self.left.assign(**{name: self.left[name].astype(typ)})
+                self.left = self.left.copy()
+                self.left[name] = self.left[name].astype(typ)
             if name in self.right.columns:
                 typ = rk.categories.dtype if rk_is_cat else object
-                self.right = self.right.assign(**{name: self.right[name].astype(typ)})
+                self.right = self.right.copy()
+                self.right[name] = self.right[name].astype(typ)
 
     def _create_cross_configuration(
         self, left: DataFrame, right: DataFrame
diff --git a/pandas/tests/reshape/merge/test_merge.py b/pandas/tests/reshape/merge/test_merge.py
index 3462fa486d936..e42dcfbe38931 100644
--- a/pandas/tests/reshape/merge/test_merge.py
+++ b/pandas/tests/reshape/merge/test_merge.py
@@ -1695,6 +1695,39 @@ def test_merge_bool_dtype(self, how, expected_data):
         expected = DataFrame(expected_data, columns=["A", "B", "C"])
         tm.assert_frame_equal(result, expected)
 
+    def test_merge_ea_with_string(self, join_type, string_dtype):
+        # GH 43734 Avoid the use of `assign` with multi-index
+        df1 = DataFrame(
+            data={
+                ("lvl0", "lvl1-a"): ["1", "2", "3", "4", None],
+                ("lvl0", "lvl1-b"): ["4", "5", "6", "7", "8"],
+            },
+            dtype=pd.StringDtype(),
+        )
+        df1_copy = df1.copy()
+        df2 = DataFrame(
+            data={
+                ("lvl0", "lvl1-a"): ["1", "2", "3", pd.NA, "5"],
+                ("lvl0", "lvl1-c"): ["7", "8", "9", pd.NA, "11"],
+            },
+            dtype=string_dtype,
+        )
+        df2_copy = df2.copy()
+        merged = merge(left=df1, right=df2, on=[("lvl0", "lvl1-a")], how=join_type)
+
+        # No change in df1 and df2
+        tm.assert_frame_equal(df1, df1_copy)
+        tm.assert_frame_equal(df2, df2_copy)
+
+        # Check the expected types for the merged data frame
+        expected = Series(
+            [np.dtype("O"), pd.StringDtype(), np.dtype("O")],
+            index=MultiIndex.from_tuples(
+                [("lvl0", "lvl1-a"), ("lvl0", "lvl1-b"), ("lvl0", "lvl1-c")]
+            ),
+        )
+        tm.assert_series_equal(merged.dtypes, expected)
+
 
 @pytest.fixture
 def left():