In my experience it is extremely common to say, for example, rex Romanus instead of rex Romae. In fact, I do not recall ever seeing a genitive when a local adjective can be used. Translating to English, this means that the Romans would much rather say "Roman king" than "king of Rome". This kind of preference varies from language to language.
In Finland we have two separate entities called The Academy of Finland and The Finnish Academy (of Science and Letters). To keep these two clearly distinct in Latin, it seems that I should call them Academia Finlandiae and Academia Fennica (scientiae litterarumque). In my opinion it is not enough for the distinction to call one of them Academia Fennica and the other one Academia Fennica scientiae litterarumque; there should be a difference in the key part of the name. But is there any classical precedent for a name like Academia Finlandiae as opposed to Academia Fennica?
Are there examples in classical Latin where an expression like rex Romae is used where rex Romanus would also be possible? The noun does not have to be rex and the adjective or genitive can be related to any place. I am just looking for examples of the general phenomenon — if there are any. If you think there are no such examples, explaining why that is the case is a good answer.
The preference of adjectives to genitives goes beyond places (see e.g. this question for a discussion of avis noctis and avis nocturna), but I want to restrict my attention to locations here.