BUG: Fix Index.equals with string vs object dtype (pandas-dev#61099)

Author: MayurKishorKumar

pandas/core/indexes/base.py

@@ -5471,12 +5471,11 @@ def equals(self, other: Any) -> bool:
             # quickly return if the lengths are different
             return False
 
-        if (isinstance(self.dtype, StringDtype) or is_object_dtype(self.dtype)) and (
-            isinstance(other.dtype, StringDtype) or is_object_dtype(other.dtype)
-        ):
-            return array_equivalent(
-                self.astype("object")._values, other.astype("object")._values
-            )
+        if (
+            (isinstance(self.dtype, StringDtype) or is_object_dtype(self.dtype)) and
+            (isinstance(other.dtype, StringDtype) or is_object_dtype(other.dtype))
+            ):
+            return array_equivalent(self._values, other._values)
 
         if (
             isinstance(self.dtype, StringDtype)