diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index 0df5a70d87655..ef68bce1d56b1 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -787,6 +787,7 @@ Reshaping
 - Bug in :meth:`DataFrame.pivot_table` incorrectly subaggregating results when called without an ``index`` argument (:issue:`58722`)
 - Bug in :meth:`DataFrame.stack` with the new implementation where ``ValueError`` is raised when ``level=[]`` (:issue:`60740`)
 - Bug in :meth:`DataFrame.unstack` producing incorrect results when manipulating empty :class:`DataFrame` with an :class:`ExtentionDtype` (:issue:`59123`)
+- Bug in :meth:`concat` where concatenating DataFrame and Series with ``ignore_index = True`` drops the series name (:issue:`60723`, :issue:`56257`)
 
 Sparse
 ^^^^^^
diff --git a/pandas/core/reshape/concat.py b/pandas/core/reshape/concat.py
index e7cb7069bbc26..5efaf0dc051bd 100644
--- a/pandas/core/reshape/concat.py
+++ b/pandas/core/reshape/concat.py
@@ -477,18 +477,23 @@ def _sanitize_mixed_ndim(
 
         else:
             name = getattr(obj, "name", None)
+            rename_columns = False
             if ignore_index or name is None:
                 if axis == 1:
                     # doing a row-wise concatenation so need everything
                     # to line up
-                    name = 0
+                    if name is None:
+                        name = 0
+                        rename_columns = True
                 else:
                     # doing a column-wise concatenation so need series
                     # to have unique names
-                    name = current_column
-                    current_column += 1
+                    if name is None:
+                        rename_columns = True
+                        name = current_column
+                        current_column += 1
                 obj = sample._constructor(obj, copy=False)
-                if isinstance(obj, ABCDataFrame):
+                if isinstance(obj, ABCDataFrame) and rename_columns:
                     obj.columns = range(name, name + 1, 1)
             else:
                 obj = sample._constructor({name: obj}, copy=False)
diff --git a/pandas/tests/reshape/concat/test_concat.py b/pandas/tests/reshape/concat/test_concat.py
index d3edee17366f7..2d0eb5d14a1d9 100644
--- a/pandas/tests/reshape/concat/test_concat.py
+++ b/pandas/tests/reshape/concat/test_concat.py
@@ -326,6 +326,8 @@ def test_concat_mixed_objs_index(self):
     def test_concat_mixed_objs_index_names(self):
         # Test row-wise concat for mixed series/frames with distinct names
         # GH2385, GH15047
+        # GH #60723 & GH #56257 (Updated the test case,
+        # as the above GH PR ones were incorrect)
 
         index = date_range("01-Jan-2013", periods=10, freq="h")
         arr = np.arange(10, dtype="int64")
@@ -341,8 +343,11 @@ def test_concat_mixed_objs_index_names(self):
         result = concat([s1, df, s2])
         tm.assert_frame_equal(result, expected)
 
-        # Rename all series to 0 when ignore_index=True
-        expected = DataFrame(np.tile(arr, 3).reshape(-1, 1), columns=[0])
+        expected = DataFrame(
+            np.kron(np.where(np.identity(3) == 1, 1, np.nan), arr).T,
+            index=np.arange(30, dtype=np.int64),
+            columns=["foo", 0, "bar"],
+        )
         result = concat([s1, df, s2], ignore_index=True)
         tm.assert_frame_equal(result, expected)
 
@@ -943,3 +948,56 @@ def test_concat_with_moot_ignore_index_and_keys():
     msg = f"Cannot set {ignore_index=} and specify keys. Either should be used."
     with pytest.raises(ValueError, match=msg):
         concat([df1, df2], keys=keys, ignore_index=ignore_index)
+
+
+@pytest.mark.parametrize(
+    "inputs, ignore_index, axis, expected",
+    [
+        # Concatenating DataFrame and named Series without ignore_index
+        (
+            [DataFrame({"a": [0, 1], "b": [2, 3]}), Series([4, 5], name="c")],
+            False,
+            0,
+            DataFrame(
+                {
+                    "a": [0, 1, None, None],
+                    "b": [2, 3, None, None],
+                    "c": [None, None, 4, 5],
+                },
+                index=[0, 1, 0, 1],
+            ),
+        ),
+        # Concatenating DataFrame and named Series with ignore_index
+        (
+            [DataFrame({"a": [0, 1], "b": [2, 3]}), Series([4, 5], name="c")],
+            True,
+            0,
+            DataFrame(
+                {
+                    "a": [0, 1, None, None],
+                    "b": [2, 3, None, None],
+                    "c": [None, None, 4, 5],
+                },
+                index=[0, 1, 2, 3],
+            ),
+        ),
+        # Concatenating DataFrame and unnamed Series along columns
+        (
+            [DataFrame({"a": [0, 1], "b": [2, 3]}), Series([4, 5]), Series([4, 5])],
+            False,
+            1,
+            DataFrame({"a": [0, 1], "b": [2, 3], 0: [4, 5], 1: [4, 5]}, index=[0, 1]),
+        ),
+        # Concatenating DataFrame and unnamed Series along columns with ignore_index
+        (
+            [DataFrame({"a": [0, 1], "b": [2, 3]}), Series([4, 5]), Series([4, 5])],
+            True,
+            1,
+            DataFrame({0: [0, 1], 1: [2, 3], 2: [4, 5], 3: [4, 5]}, index=[0, 1]),
+        ),
+    ],
+)
+def test_concat_of_series_and_frame(inputs, ignore_index, axis, expected):
+    # GH #60723 and #56257
+    result = concat(inputs, ignore_index=ignore_index, axis=axis)
+    tm.assert_frame_equal(result, expected)