diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 6219aa07478d7..7b58e34152547 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -3077,7 +3077,7 @@ def intersection(self, other, sort=False):
             return Index([], name=result_name)
 
         elif not is_dtype_equal(self.dtype, other.dtype):
-            dtype = find_common_type([self.dtype, other.dtype])
+            dtype = self._find_common_type_compat(other)
             this = self.astype(dtype, copy=False)
             other = other.astype(dtype, copy=False)
             return this.intersection(other, sort=sort)
@@ -3429,6 +3429,10 @@ def get_indexer(
                 ptarget, method=method, limit=limit, tolerance=tolerance
             )
 
+        if is_dtype_equal(self.dtype, target.dtype) and self.equals(target):
+            # Only call equals if we have same dtype to avoid inference/casting
+            return np.arange(len(target), dtype=np.intp)
+
         return self._get_indexer(target, method, limit, tolerance)
 
     def _get_indexer(
@@ -3937,8 +3941,9 @@ def join(
             return join_index, lidx, ridx
 
         if not is_dtype_equal(self.dtype, other.dtype):
-            this = self.astype("O")
-            other = other.astype("O")
+            dtype = self._find_common_type_compat(other)
+            this = self.astype(dtype, copy=False)
+            other = other.astype(dtype, copy=False)
             return this.join(other, how=how, return_indexers=True)
 
         _validate_join_method(how)
@@ -5216,6 +5221,8 @@ def get_indexer_non_unique(self, target) -> tuple[np.ndarray, np.ndarray]:
             that = target.astype(dtype, copy=False)
             return this.get_indexer_non_unique(that)
 
+        # Note: _maybe_promote ensures we never get here with MultiIndex
+        #  self and non-Multi target
         tgt_values = target._get_engine_target()
 
         indexer, missing = self._engine.get_indexer_non_unique(tgt_values)
@@ -5948,8 +5955,7 @@ def insert(self, loc: int, item) -> Index:
         try:
             item = self._validate_fill_value(item)
         except TypeError:
-            inferred, _ = infer_dtype_from(item)
-            dtype = find_common_type([self.dtype, inferred])
+            dtype = self._find_common_type_compat(item)
             return self.astype(dtype).insert(loc, item)
 
         arr = np.asarray(self)
diff --git a/pandas/core/indexes/interval.py b/pandas/core/indexes/interval.py
index e71d6d690bcda..e6764dd01e6d3 100644
--- a/pandas/core/indexes/interval.py
+++ b/pandas/core/indexes/interval.py
@@ -646,10 +646,6 @@ def _get_indexer(
         # returned ndarray is np.intp
 
         if isinstance(target, IntervalIndex):
-            # equal indexes -> 1:1 positional match
-            if self.equals(target):
-                return np.arange(len(self), dtype="intp")
-
             if not self._should_compare(target):
                 return self._get_indexer_non_comparable(target, method, unique=True)
 
diff --git a/pandas/tests/libs/test_hashtable.py b/pandas/tests/libs/test_hashtable.py
index 08bfc74e0ef8d..5ff20051da8c0 100644
--- a/pandas/tests/libs/test_hashtable.py
+++ b/pandas/tests/libs/test_hashtable.py
@@ -252,9 +252,9 @@ def test_get_labels_groupby_for_Int64(writable):
     vals = np.array([1, 2, -1, 2, 1, -1], dtype=np.int64)
     vals.flags.writeable = writable
     arr, unique = table.get_labels_groupby(vals)
-    expected_arr = np.array([0, 1, -1, 1, 0, -1], dtype=np.int64)
+    expected_arr = np.array([0, 1, -1, 1, 0, -1], dtype=np.intp)
     expected_unique = np.array([1, 2], dtype=np.int64)
-    tm.assert_numpy_array_equal(arr.astype(np.int64), expected_arr)
+    tm.assert_numpy_array_equal(arr, expected_arr)
     tm.assert_numpy_array_equal(unique, expected_unique)