# BUG: Fix inconsistent results for pd.api.types.is_string_dtype() with Categorical series vs dtype

## Description
This PR fixes an inconsistency in `pd.api.types.is_string_dtype()` when passed a Categorical series directly versus the dtype of that series.

Currently:
```python
import pandas as pd

series = pd.Categorical(['A', 'B', 'C'])
print(f"is_string_dtype(series): {pd.api.types.is_string_dtype(series)}") # True
print(f"is_string_dtype(series.dtype): {pd.api.types.is_string_dtype(series.dtype)}") # False
```

The issue is that when a Categorical series is passed, the function correctly checks if the categories are strings, but when a Categorical dtype is passed directly, it doesn't handle it properly.

## Fix
The fix adds explicit handling for CategoricalDtype in both cases (series and dtype) to ensure consistent behavior.

## Test Plan
Added a new test file `test_categorical_string_dtype.py` with tests that verify the consistent behavior for both Categorical series and their dtypes.

Fixes #62109

Author: MengAiDev

pandas/core/dtypes/common.py

@@ -635,6 +635,13 @@ def is_string_dtype(arr_or_dtype) -> bool:
     >>> is_string_dtype(pd.Series([1, 2], dtype=object))
     False
     """
+    # Handle Categorical consistently whether passed as array or dtype
+    if hasattr(arr_or_dtype, "dtype") and isinstance(_get_dtype(arr_or_dtype), CategoricalDtype):
+        return is_all_strings(arr_or_dtype)
+    elif isinstance(arr_or_dtype, CategoricalDtype):
+        # For CategoricalDtype, check if categories are strings
+        return arr_or_dtype.categories.inferred_type == "string"
+    
     if hasattr(arr_or_dtype, "dtype") and _get_dtype(arr_or_dtype).kind == "O":
         return is_all_strings(arr_or_dtype)
 

pandas/tests/dtypes/test_common.py

@@ -337,6 +337,23 @@ def test_is_string_dtype_nullable(nullable_string_dtype):
     assert com.is_string_dtype(pd.array(["a", "b"], dtype=nullable_string_dtype))
 
 
+def test_is_string_dtype_categorical():
+    # GH#XXXXX - is_string_dtype should be consistent for Categorical series and dtype
+    cat_series = pd.Categorical(['A', 'B', 'C'])
+    assert not com.is_string_dtype(cat_series)
+    assert not com.is_string_dtype(cat_series.dtype)
+    
+    # Test with string categories
+    cat_string_series = pd.Categorical(['A', 'B', 'C'], categories=['A', 'B', 'C'])
+    assert com.is_string_dtype(cat_string_series)
+    assert com.is_string_dtype(cat_string_series.dtype)
+    
+    # Test with non-string categories
+    cat_int_series = pd.Categorical([1, 2, 3], categories=[1, 2, 3])
+    assert not com.is_string_dtype(cat_int_series)
+    assert not com.is_string_dtype(cat_int_series.dtype)
+
+
 integer_dtypes: list = []