diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index 5601048c409e1..492df4f1e3612 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -535,7 +535,7 @@ Indexing
 - Bug in :meth:`DataFrame.sort_index` where ``ignore_index=True`` was not being respected when the index was already sorted (:issue:`43591`)
 - Bug in :meth:`Index.get_indexer_non_unique` when index contains multiple ``np.datetime64("NaT")`` and ``np.timedelta64("NaT")`` (:issue:`43869`)
 - Bug in setting a scalar :class:`Interval` value into a :class:`Series` with ``IntervalDtype`` when the scalar's sides are floats and the values' sides are integers (:issue:`44201`)
--
+- Bug when setting string-backed :class:`Categorical` values that can be parsed to datetimes into a :class:`DatetimeArray` or :class:`Series` or :class:`DataFrame` column backed by :class:`DatetimeArray` failing to parse these strings (:issue:`44236`)
 
 
 Missing
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 72c00dfe7c65a..f8aa1656c8c30 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -68,6 +68,7 @@
 from pandas.util._exceptions import find_stack_level
 
 from pandas.core.dtypes.common import (
+    is_all_strings,
     is_categorical_dtype,
     is_datetime64_any_dtype,
     is_datetime64_dtype,
@@ -720,7 +721,7 @@ def _validate_listlike(self, value, allow_object: bool = False):
         value = pd_array(value)
         value = extract_array(value, extract_numpy=True)
 
-        if is_dtype_equal(value.dtype, "string"):
+        if is_all_strings(value):
             # We got a StringArray
             try:
                 # TODO: Could use from_sequence_of_strings if implemented
diff --git a/pandas/core/dtypes/common.py b/pandas/core/dtypes/common.py
index 0788ecdd8b4b5..815a0a2040ddb 100644
--- a/pandas/core/dtypes/common.py
+++ b/pandas/core/dtypes/common.py
@@ -15,6 +15,7 @@
     Interval,
     Period,
     algos,
+    lib,
 )
 from pandas._libs.tslibs import conversion
 from pandas._typing import (
@@ -1788,3 +1789,23 @@ def pandas_dtype(dtype) -> DtypeObj:
         raise TypeError(f"dtype '{dtype}' not understood")
 
     return npdtype
+
+
+def is_all_strings(value: ArrayLike) -> bool:
+    """
+    Check if this is an array of strings that we should try parsing.
+
+    Includes object-dtype ndarray containing all-strings, StringArray,
+    and Categorical with all-string categories.
+    Does not include numpy string dtypes.
+    """
+    dtype = value.dtype
+
+    if isinstance(dtype, np.dtype):
+        return (
+            dtype == np.dtype("object")
+            and lib.infer_dtype(value, skipna=False) == "string"
+        )
+    elif isinstance(dtype, CategoricalDtype):
+        return dtype.categories.inferred_type == "string"
+    return dtype == "string"
diff --git a/pandas/tests/indexing/test_indexing.py b/pandas/tests/indexing/test_indexing.py
index 4604fad019eca..d6402e027be98 100644
--- a/pandas/tests/indexing/test_indexing.py
+++ b/pandas/tests/indexing/test_indexing.py
@@ -871,7 +871,7 @@ def test_setitem_dt64_string_scalar(self, tz_naive_fixture, indexer_sli):
         else:
             assert ser._values is values
 
-    @pytest.mark.parametrize("box", [list, np.array, pd.array])
+    @pytest.mark.parametrize("box", [list, np.array, pd.array, pd.Categorical, Index])
     @pytest.mark.parametrize(
         "key", [[0, 1], slice(0, 2), np.array([True, True, False])]
     )
@@ -911,7 +911,7 @@ def test_setitem_td64_scalar(self, indexer_sli, scalar):
         indexer_sli(ser)[0] = scalar
         assert ser._values._data is values._data
 
-    @pytest.mark.parametrize("box", [list, np.array, pd.array])
+    @pytest.mark.parametrize("box", [list, np.array, pd.array, pd.Categorical, Index])
     @pytest.mark.parametrize(
         "key", [[0, 1], slice(0, 2), np.array([True, True, False])]
     )