REF: Simplify StringDtype check to allow fallthrough

Author: zacharym-collins

pandas/io/pytables.py

@@ -3147,14 +3147,12 @@ def write_multi_index(self, key: str, index: MultiIndex) -> None:
             zip(index.levels, index.codes, index.names, strict=True)
         ):
             # write the level
-            if isinstance(lev.dtype, ExtensionDtype):
-                # GH 63412
-                if isinstance(lev.dtype, StringDtype):
-                    pass
-                else:
-                    raise NotImplementedError(
-                        "Saving a MultiIndex with an extension dtype is not supported."
-                    )
+            if isinstance(lev.dtype, ExtensionDtype) and not isinstance(
+                lev.dtype, StringDtype
+            ):
+                raise NotImplementedError(
+                    "Saving a MultiIndex with an extension dtype is not supported."
+                )
             level_key = f"{key}_level{i}"
             conv_level = _convert_index(level_key, lev, self.encoding, self.errors)
             self.write_array(level_key, conv_level.values)