diff --git a/pandas/_testing/asserters.py b/pandas/_testing/asserters.py
index 047e8f91df23e..bbd5e60a5a812 100644
--- a/pandas/_testing/asserters.py
+++ b/pandas/_testing/asserters.py
@@ -188,7 +188,7 @@ def assert_index_equal(
     check_order: bool = True,
     rtol: float = 1.0e-5,
     atol: float = 1.0e-8,
-    obj: str = "Index",
+    obj: str | None = None,
 ) -> None:
     """
     Check that left and right Index are equal.
@@ -217,7 +217,7 @@ def assert_index_equal(
         Relative tolerance. Only used when check_exact is False.
     atol : float, default 1e-8
         Absolute tolerance. Only used when check_exact is False.
-    obj : str, default 'Index'
+    obj : str, default 'Index' or 'MultiIndex'
         Specify object name being compared, internally used to show appropriate
         assertion message.
 
@@ -235,6 +235,9 @@ def assert_index_equal(
     """
     __tracebackhide__ = True
 
+    if obj is None:
+        obj = "MultiIndex" if isinstance(left, MultiIndex) else "Index"
+
     def _check_types(left, right, obj: str = "Index") -> None:
         if not exact:
             return
@@ -283,7 +286,7 @@ def _check_types(left, right, obj: str = "Index") -> None:
         right = cast(MultiIndex, right)
 
         for level in range(left.nlevels):
-            lobj = f"MultiIndex level [{level}]"
+            lobj = f"{obj} level [{level}]"
             try:
                 # try comparison on levels/codes to avoid densifying MultiIndex
                 assert_index_equal(
@@ -314,7 +317,7 @@ def _check_types(left, right, obj: str = "Index") -> None:
                     obj=lobj,
                 )
             # get_level_values may change dtype
-            _check_types(left.levels[level], right.levels[level], obj=obj)
+            _check_types(left.levels[level], right.levels[level], obj=lobj)
 
     # skip exact index checking when `check_categorical` is False
     elif check_exact and check_categorical:
@@ -527,7 +530,7 @@ def assert_interval_array_equal(
         kwargs["check_freq"] = False
 
     assert_equal(left._left, right._left, obj=f"{obj}.left", **kwargs)
-    assert_equal(left._right, right._right, obj=f"{obj}.left", **kwargs)
+    assert_equal(left._right, right._right, obj=f"{obj}.right", **kwargs)
 
     assert_attr_equal("closed", left, right, obj=obj)
 
diff --git a/pandas/tests/util/test_assert_frame_equal.py b/pandas/tests/util/test_assert_frame_equal.py
index 28d96ea25cba7..ea954756d63c8 100644
--- a/pandas/tests/util/test_assert_frame_equal.py
+++ b/pandas/tests/util/test_assert_frame_equal.py
@@ -79,7 +79,7 @@ def test_frame_equal_shape_mismatch(df1, df2, frame_or_series):
             DataFrame.from_records(
                 {"a": [1.0, 2.0], "b": [2.1, 1.5], "c": ["l1", "l2"]}, index=["a", "b"]
             ),
-            "MultiIndex level \\[0\\] are different",
+            "DataFrame\\.index level \\[0\\] are different",
         ),
     ],
 )
diff --git a/pandas/tests/util/test_assert_interval_array_equal.py b/pandas/tests/util/test_assert_interval_array_equal.py
index 8cc4ade3d7e95..aad27672c0f6f 100644
--- a/pandas/tests/util/test_assert_interval_array_equal.py
+++ b/pandas/tests/util/test_assert_interval_array_equal.py
@@ -1,7 +1,11 @@
 import pytest
 
-from pandas import interval_range
+from pandas import (
+    Interval,
+    interval_range,
+)
 import pandas._testing as tm
+from pandas.arrays import IntervalArray
 
 
 @pytest.mark.parametrize(
@@ -79,3 +83,18 @@ def test_interval_array_equal_start_mismatch():
 
     with pytest.raises(AssertionError, match=msg):
         tm.assert_interval_array_equal(arr1, arr2)
+
+
+def test_interval_array_equal_end_mismatch_only():
+    arr1 = IntervalArray([Interval(0, 1), Interval(0, 5)])
+    arr2 = IntervalArray([Interval(0, 1), Interval(0, 6)])
+
+    msg = """\
+IntervalArray.right are different
+
+IntervalArray.right values are different \\(50.0 %\\)
+\\[left\\]:  \\[1, 5\\]
+\\[right\\]: \\[1, 6\\]"""
+
+    with pytest.raises(AssertionError, match=msg):
+        tm.assert_interval_array_equal(arr1, arr2)
diff --git a/pandas/tests/util/test_assert_series_equal.py b/pandas/tests/util/test_assert_series_equal.py
index a3b24c029fbac..c3cd90f2edfb3 100644
--- a/pandas/tests/util/test_assert_series_equal.py
+++ b/pandas/tests/util/test_assert_series_equal.py
@@ -137,7 +137,7 @@ def test_less_precise(data1, data2, any_float_dtype, decimals):
             DataFrame.from_records(
                 {"a": [1.0, 2.0], "b": [2.1, 1.5], "c": ["l1", "l2"]}, index=["a", "b"]
             ).c,
-            "MultiIndex level \\[0\\] are different",
+            "Series\\.index level \\[0\\] are different",
         ),
     ],
 )