diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 11533647ca124..36a24c93600b8 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -432,6 +432,8 @@ Groupby/resample/rolling
 Reshaping
 ^^^^^^^^^
 - Bug in :func:`crosstab` when ``dropna=False`` would not keep ``np.nan`` in the result (:issue:`10772`)
+- Bug in :meth:`BlockManager.insert` allows assignments for multiple-dimension ndarray into one column (:issue:`53366`)
+- Bug in :meth:`DataFrame.__setitem__` allows assignments for multiple-dimension ndarray into one column (:issue:`51925`)
 - Bug in :meth:`DataFrame.agg` and :meth:`Series.agg` on non-unique columns would return incorrect type when dist-like argument passed in (:issue:`51099`)
 - Bug in :meth:`DataFrame.idxmin` and :meth:`DataFrame.idxmax`, where the axis dtype would be lost for empty frames (:issue:`53265`)
 - Bug in :meth:`DataFrame.merge` not merging correctly when having ``MultiIndex`` with single level (:issue:`52331`)
diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 8afb3ee96ba94..59282f2e8fe12 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -3966,6 +3966,24 @@ def __setitem__(self, key, value):
         ):
             # Column to set is duplicated
             self._setitem_array([key], value)
+        elif (
+            isinstance(value, np.ndarray) and value.ndim > 1 and self.columns.is_unique
+        ):
+            # TODO: a check for MultiIndex should be added
+            if isinstance(self.columns, MultiIndex):
+                self._set_item(key, value)
+                return
+            # squeeze 2d ndarray to 1d if possible
+            # this keeps the backward compatibility
+            if np.ndim(value) == 2 and (1 in np.shape(value)):
+                # if value is np.matrix, convert to np.ndarray
+                value = np.asarray(value).flatten()
+                self._set_item(key, value)
+            else:
+                # avoid assign non 1d array to column
+                raise ValueError(
+                    f"Expected a 1D array, got an array with shape {value.shape}"
+                )
         else:
             # set column
             self._set_item(key, value)
diff --git a/pandas/core/internals/managers.py b/pandas/core/internals/managers.py
index 2a7c0536c66a4..b5cbde6dbd65c 100644
--- a/pandas/core/internals/managers.py
+++ b/pandas/core/internals/managers.py
@@ -1401,7 +1401,7 @@ def insert(self, loc: int, item: Hashable, value: ArrayLike, refs=None) -> None:
         # insert to the axis; this could possibly raise a TypeError
         new_axis = self.items.insert(loc, item)
 
-        if value.ndim == 2:
+        if value.ndim >= 2:
             value = value.T
             if len(value) > 1:
                 raise ValueError(
diff --git a/pandas/tests/frame/test_constructors.py b/pandas/tests/frame/test_constructors.py
index 47e307f561cf4..a011da84804b2 100644
--- a/pandas/tests/frame/test_constructors.py
+++ b/pandas/tests/frame/test_constructors.py
@@ -283,6 +283,11 @@ def test_constructor_cast_failure(self):
         with pytest.raises(ValueError, match=msg):
             df["test"] = np.ones((4, 2))
 
+        # this is not ok
+        msg = "Expected a 1D array, got an array with shape \\(4, 2, 3\\)"
+        with pytest.raises(ValueError, match=msg):
+            df["test"] = np.ones((4, 2, 3))
+
         # this is ok
         df["foo2"] = np.ones((4, 2)).tolist()