I fixed and removed that duplicated line of code and the tests i wrote passed.
Though it seems another test broke: test_run_error_in_application, only when the error is simulated inside application.initialize()

And for what I understand it's really not an issue, since at this point of the execution the updater hasn't been initialized yet, and that's why app.shutdown() doesn't call updater.shutdown()

If my assumptions are correct, i may fix the test asserting app._initialized and app.updater._initilized as false, removing the need to monkeypatch the methods
Alternatively i can assert the two different cases inside a if statement
Something like assert set(shutdowns) == ( {"application", "updater"} if method == "start" else {"application"} )

Edit: I've tested this second option and i can commit it as soon i as i have a green light :)

Edit 2: I think that what i wrote before was entirely wrong, so i leave it up to you big guys to help me understand where the issue is and I'll try and fix it for you

The bad boy is this here:

this tells Application.shutdown to just exit if the app wasn't properly initialized and hence Updater.shutdown is not called.
While we can't be 100% sure that Updater.initialize wasn't called by Application.initialize, I'm okay with that. Catching all errors that can happen during startup is a kind of a sisyphus task, so a best effort suffices IMO.

Thank you for the contribution :)