API: Allow non-tuples in pandas.merge

Closes pandas-dev#34741, while
retaining the spirit of the spirit of
pandas-dev#34208.

Author: TomAugspurger

doc/source/whatsnew/v1.1.0.rst

@@ -746,6 +746,7 @@ Deprecations
 - :meth:`DataFrame.to_dict` has deprecated accepting short names for ``orient`` in future versions (:issue:`32515`)
 - :meth:`Categorical.to_dense` is deprecated and will be removed in a future version, use ``np.asarray(cat)`` instead (:issue:`32639`)
 - The ``fastpath`` keyword in the ``SingleBlockManager`` constructor is deprecated and will be removed in a future version (:issue:`33092`)
+- Providing ``suffixes`` as a ``set`` in :func:`pandas.merge` is deprecated. Provide a tuple instead (:issue:`33740`, `34741`).
 - :meth:`Index.is_mixed` is deprecated and will be removed in a future version, check ``index.inferred_type`` directly instead (:issue:`32922`)
 
 - Passing any arguments but the first one to  :func:`read_html` as

pandas/core/reshape/merge.py

@@ -2072,9 +2072,13 @@ def _items_overlap_with_suffix(left: Index, right: Index, suffixes: Tuple[str, s
     If corresponding suffix is empty, the entry is simply converted to string.
 
     """
-    if not isinstance(suffixes, tuple):
-        raise TypeError(
-            f"suffixes should be tuple of (str, str). But got {type(suffixes).__name__}"
+    if isinstance(suffixes, set):
+        warnings.warn(
+            "Passing 'suffixes' as a set, which is unordered, may result in "
+            "unexpected results. Provide 'suffixes' as a tuple instead. In the "
+            "future a 'TypeError' will be raised.",
+            FutureWarning,
+            stacklevel=4,
         )
 
     to_rename = left.intersection(right)

pandas/tests/reshape/merge/test_merge.py

@@ -2069,18 +2069,12 @@ def test_merge_suffix_error(col1, col2, suffixes):
         pd.merge(a, b, left_index=True, right_index=True, suffixes=suffixes)
 
 
-@pytest.mark.parametrize(
-    "col1, col2, suffixes", [("a", "a", {"a", "b"}), ("a", "a", None), (0, 0, None)],
-)
-def test_merge_suffix_type_error(col1, col2, suffixes):
-    a = pd.DataFrame({col1: [1, 2, 3]})
-    b = pd.DataFrame({col2: [3, 4, 5]})
+def test_merge_suffix_set():
+    a = pd.DataFrame({"a": [1, 2, 3]})
+    b = pd.DataFrame({"b": [3, 4, 5]})
 
-    msg = (
-        f"suffixes should be tuple of \\(str, str\\). But got {type(suffixes).__name__}"
-    )
-    with pytest.raises(TypeError, match=msg):
-        pd.merge(a, b, left_index=True, right_index=True, suffixes=suffixes)
+    with tm.assert_produces_warning(FutureWarning):
+        pd.merge(a, b, left_index=True, right_index=True, suffixes={"left", "right"})
 
 
 @pytest.mark.parametrize(