Maybe I'm reading things into what you write, but my impression is that it is there, even if in an attenuated, rejected form. As the what it should be called - I don't know. Presumably at least as a child you felt Hungarian to some extent as the default, almost automatic way children assume things. That wouldn't necessarily been the case with Germanness as a foreigner, though it could have been (I have no way of knowing).

In my case, while I developed a strong feeling of attachment to Geneva and to French culture in the broadest sense of the word, to the extent of feeling a sense of being at home when I visited Quebec in college, I never got any Swiss or French national identity - I was always a foreigner.