There are a few odd cases, some of which have been mentioned, but I think the main use case for that are actually refugee travel documents (based on the 1951 convention or otherwise) and 1954 convention travel documents for stateless people. Technically they are sometimes called “travel documents” instead of “passport” but they look like passports, are used like passports and you would need to fill in this form if you had one.
Now, by definition, stateless people don't have a citizenship and refugees cancannot avail themselves of the protection of their country of citizenship (and in particular they cannot apply for a passport, even if they are still citizens) so in both cases the travel document is issued by the state where they reside, which is not their country of citizenship.