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The dictionary and the set do not give the freedom to add dynamic attributes to them. I agree that the Counter should have the same behaviour.
However, this will raise the same bug when we inherit from a Counter object.
>>> class mylist(list): pass
... 
>>> l = mylist()
>>> l.foo = "bar"
>>> c = copy.deepcopy(l)
>>> print(c.foo) # prints bar
>>> class myCounter(Counter): pass
... 
>>> original = myCounter()
>>> original.foo = "bar"
>>> c = copy.deepcopy(original)
>>> c.foo
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
AttributeError: 'myCounter' object has no attribute 'foo'
The reduction function should still copy every dynamic attribute of the object.