diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index e19021762792f..9d9eee5070377 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -367,7 +367,7 @@ def _get_axis_name(cls, axis):
                 return cls._AXIS_NAMES[axis]
             except KeyError:
                 pass
-        raise ValueError(f"No axis named {axis} for object type {cls}")
+        raise ValueError(f"No axis named {axis} for object type {cls.__name__}")
 
     def _get_axis(self, axis):
         name = self._get_axis_name(axis)
diff --git a/pandas/tests/dtypes/test_dtypes.py b/pandas/tests/dtypes/test_dtypes.py
index 16ee7c27780ca..658d27160e3e1 100644
--- a/pandas/tests/dtypes/test_dtypes.py
+++ b/pandas/tests/dtypes/test_dtypes.py
@@ -479,7 +479,6 @@ def test_basic(self, dtype):
 
     def test_empty(self):
         dt = PeriodDtype()
-        # https://github.com/pandas-dev/pandas/issues/27388
         msg = "object has no attribute 'freqstr'"
         with pytest.raises(AttributeError, match=msg):
             str(dt)
diff --git a/pandas/tests/frame/methods/test_to_period.py b/pandas/tests/frame/methods/test_to_period.py
index eac78e611b008..051461b6c554d 100644
--- a/pandas/tests/frame/methods/test_to_period.py
+++ b/pandas/tests/frame/methods/test_to_period.py
@@ -31,6 +31,6 @@ def test_frame_to_period(self):
         pts = df.to_period("M", axis=1)
         tm.assert_index_equal(pts.columns, exp.columns.asfreq("M"))
 
-        msg = "No axis named 2 for object type <class 'pandas.core.frame.DataFrame'>"
+        msg = "No axis named 2 for object type DataFrame"
         with pytest.raises(ValueError, match=msg):
             df.to_period(axis=2)
diff --git a/pandas/tests/generic/test_series.py b/pandas/tests/generic/test_series.py
index bfdfd6d319b3f..229c6782e995e 100644
--- a/pandas/tests/generic/test_series.py
+++ b/pandas/tests/generic/test_series.py
@@ -57,7 +57,8 @@ def test_set_axis_name_mi(self, func):
 
     def test_set_axis_name_raises(self):
         s = pd.Series([1])
-        with pytest.raises(ValueError):
+        msg = "No axis named 1 for object type Series"
+        with pytest.raises(ValueError, match=msg):
             s._set_axis_name(name="a", axis=1)
 
     def test_get_numeric_data_preserve_dtype(self):