This looks great! Ideally, though, there are also tests to ensure we don't regress... I think it can be fairly simple, along the lines of

class TestOverride:
    @classmethod
    def setup_class(cls):
        class Override:
            def __array_function__(self, *args, **kwargs):
                return "function"
            def __array_ufunc__(self, *args, **kwargs):
                return "ufunc"

        cls.override = Override()

    @pytest.mark.parametrize("func, kwargs", [
        (np.strings.upper, {}),
        (...),
    ])
    def test_override_function(self, func, kwargs):
        assert func(self.override, **kwargs) == "function"

    ... same for any ufuncs ...

Also, I think this warrants a changelog entry...

@byrdie - the problem is that some of the functions pass the input through np.asanyarray - so the Override class gets turned into an ndarray with object dtype. I fear those functions just need an array_function_dispatch as well (and then move their test to the one using __array_function__).

Slightly annoying, but also good, in that the tests showed the PR was not quite complete yet -- you need 8 more function:

center
expandtabs
ljust
multiply
partition
replace
rjust
rpartition
zfill

Thanks, this looks all OK now! The test failures indeed seem unrelated, so approving.

Might be good, though, for @ngoldbaum to have a quick look too.

Great that everything now passes too. I'll re-approve but will give it a few more days just in case @mattip has other comments, or @ngoldbaum wants to have a look. (If nothing happens, feel free to ping me middle of next week, and I'll merge.)