diff --git a/pandas/_testing/asserters.py b/pandas/_testing/asserters.py
index 1942e07d1b562..a29767153b021 100644
--- a/pandas/_testing/asserters.py
+++ b/pandas/_testing/asserters.py
@@ -314,18 +314,16 @@ def _check_types(left, right, obj="Index") -> None:
             return
 
         assert_class_equal(left, right, exact=exact, obj=obj)
+        assert_attr_equal("inferred_type", left, right, obj=obj)
 
         # Skip exact dtype checking when `check_categorical` is False
-        if check_categorical:
-            assert_attr_equal("dtype", left, right, obj=obj)
-            if is_categorical_dtype(left.dtype) and is_categorical_dtype(right.dtype):
+        if is_categorical_dtype(left.dtype) and is_categorical_dtype(right.dtype):
+            if check_categorical:
+                assert_attr_equal("dtype", left, right, obj=obj)
                 assert_index_equal(left.categories, right.categories, exact=exact)
+            return
 
-        # allow string-like to have different inferred_types
-        if left.inferred_type in ("string"):
-            assert right.inferred_type in ("string")
-        else:
-            assert_attr_equal("inferred_type", left, right, obj=obj)
+        assert_attr_equal("dtype", left, right, obj=obj)
 
     def _get_ilevel_values(index, level):
         # accept level number only
@@ -437,6 +435,8 @@ def assert_class_equal(left, right, exact: bool | str = True, obj="Input"):
     """
     Checks classes are equal.
     """
+    from pandas.core.indexes.numeric import NumericIndex
+
     __tracebackhide__ = True
 
     def repr_class(x):
@@ -446,17 +446,16 @@ def repr_class(x):
 
         return type(x).__name__
 
+    if type(left) == type(right):
+        return
+
     if exact == "equiv":
-        if type(left) != type(right):
-            # allow equivalence of Int64Index/RangeIndex
-            types = {type(left).__name__, type(right).__name__}
-            if len(types - {"Int64Index", "RangeIndex"}):
-                msg = f"{obj} classes are not equivalent"
-                raise_assert_detail(obj, msg, repr_class(left), repr_class(right))
-    elif exact:
-        if type(left) != type(right):
-            msg = f"{obj} classes are different"
-            raise_assert_detail(obj, msg, repr_class(left), repr_class(right))
+        # accept equivalence of NumericIndex (sub-)classes
+        if isinstance(left, NumericIndex) and isinstance(right, NumericIndex):
+            return
+
+    msg = f"{obj} classes are different"
+    raise_assert_detail(obj, msg, repr_class(left), repr_class(right))
 
 
 def assert_attr_equal(attr: str, left, right, obj: str = "Attributes"):
diff --git a/pandas/tests/util/test_assert_index_equal.py b/pandas/tests/util/test_assert_index_equal.py
index 1778b6fb9d832..8211b52fed650 100644
--- a/pandas/tests/util/test_assert_index_equal.py
+++ b/pandas/tests/util/test_assert_index_equal.py
@@ -58,15 +58,30 @@ def test_index_equal_length_mismatch(check_exact):
         tm.assert_index_equal(idx1, idx2, check_exact=check_exact)
 
 
-def test_index_equal_class_mismatch(check_exact):
-    msg = """Index are different
+@pytest.mark.parametrize("exact", [False, "equiv"])
+def test_index_equal_class(exact):
+    idx1 = Index([0, 1, 2])
+    idx2 = RangeIndex(3)
+
+    tm.assert_index_equal(idx1, idx2, exact=exact)
+
+
+@pytest.mark.parametrize(
+    "idx_values, msg_str",
+    [
+        [[1, 2, 3.0], "Float64Index\\(\\[1\\.0, 2\\.0, 3\\.0\\], dtype='float64'\\)"],
+        [range(3), "RangeIndex\\(start=0, stop=3, step=1\\)"],
+    ],
+)
+def test_index_equal_class_mismatch(check_exact, idx_values, msg_str):
+    msg = f"""Index are different
 
 Index classes are different
 \\[left\\]:  Int64Index\\(\\[1, 2, 3\\], dtype='int64'\\)
-\\[right\\]: Float64Index\\(\\[1\\.0, 2\\.0, 3\\.0\\], dtype='float64'\\)"""
+\\[right\\]: {msg_str}"""
 
     idx1 = Index([1, 2, 3])
-    idx2 = Index([1, 2, 3.0])
+    idx2 = Index(idx_values)
 
     with pytest.raises(AssertionError, match=msg):
         tm.assert_index_equal(idx1, idx2, exact=True, check_exact=check_exact)