Fix false-positive of unused-private-member with class name

ChangeLog

@@ -60,6 +60,10 @@ Release date: TBA
 
   Closes #4673
 
+* Fix a false positive for ``unused-private-member`` with class names
+
+  Closes #4681
+
 
 What's New in Pylint 2.9.6?
 ===========================

pylint/checkers/classes.py

@@ -992,6 +992,7 @@ def _check_unused_private_attributes(self, node: astroid.ClassDef) -> None:
                 )
 
             for attribute in node.nodes_of_class(astroid.Attribute):
+                # pylint: disable=too-many-boolean-expressions
                 if attribute.attrname == assign_attr.attrname and (
                     (
                         # If assigned to cls.attrib, can be accessed by cls/self
@@ -1004,6 +1005,7 @@ def _check_unused_private_attributes(self, node: astroid.ClassDef) -> None:
                         assign_attr.expr.name in acceptable_obj_names
                         and attribute.expr.name == "self"
                     )
+                    or (assign_attr.expr.name == attribute.expr.name == node.name)
                 ):
                     break
             else:

tests/functional/u/unused/unused_private_member.py

@@ -201,3 +201,22 @@ def __inner_4():  # [unused-private-member]
 
         fn_to_return = __inner_1 if flag else __inner_3(__inner_2)
         return fn_to_return
+
+
+# Test cases for false-positive reported in #4681
+# https://github.com/PyCQA/pylint/issues/4681
+class FalsePositive4681:
+    __instance = None
+
+    @staticmethod
+    def instance():
+        if FalsePositive4681.__instance is None:
+            FalsePositive4681()
+        return FalsePositive4681.__instance
+
+    def __init__(self):
+        try:
+            FalsePositive4681.__instance = 42  # This should be fine
+        except Exception:  # pylint: disable=broad-except
+            print("Error")
+            FalsePositive4681.__instance = False  # This should be fine