Thanks Elizabeth for the test. The regression test seems to be same as the case noted by Serhiy that Foo.__eq__ is not returning NotImplemented so that ANY.__eq__ can be executed. Below would be the correct implementation that passes. 

The actual comparison is done at [0]. If Foo.__eq__ returned NotImplemented due to type difference it would have called other.arguments. So there is no chance for ANY.__eq__ to be executed. I feel it's more about the third party class that needs to be fixed rather than the stdlib code here and changing the order of arguments for ANY's __eq__ precedence might introduce other subtle bugs.

self.arguments = OrderedDict([('args', (<__main__.Foo object at 0x10a54b500>,))])
other.arguments = OrderedDict([('args', (<ANY>,))])

def __eq__(self, other):
    if self is other:
        return True
    if not isinstance(other, BoundArguments):
        return NotImplemented
    return (self.signature == other.signature and
            self.arguments == other.arguments)

# Better implementation

from unittest.mock import Mock, call, ANY

class Foo(object):
     def __eq__(self, other):
          if not isinstance(other, self.__class__):
               return NotImplemented
          return True
     def __ne__(self, other): pass

mock = Mock(spec_set=Foo)
expected = [call(ANY)]
mock(Foo())

mock.assert_has_calls(expected)

[0] https://github.com/python/cpython/blob/cd6e83b4810549c308ab2d7315dbab526e35ccf6/Lib/inspect.py#L2708


3.5 and 3.6 are in security fixes only mode. If this is considered to be a bug it can go in master, 3.8 and 3.7. The tests section is for CPython test suite and not for the unittest related bugs. So triaging it back.