I don't know much about grammar (I just learn languages

) but it seems strange that 'bildøra' is not a kind of possessive, when this is the normal way of saying 'la porte de la voiture', which obviously is possessive. If you take the examples with Kari (Karis bil, Kari sin bil, bilen til Kari), all would in theory be possible in Norwegian, but would be very unusual, especially the first two: bilens dør, bilen sin dør, døra til bilen. So it seems that the normal, and perhaps the only, kind of possessive in Romance languages, when translated with exactly the same meaning into Scandinavian languages, is no longer possessive (?)
In some sense, you are right: 'bildøra' could absolutely be considered some kind possessive. That is indeed the most natural and salient interpretation of the compound. The question is if compounds deserve to be labelled with such a term because compounds are also compatible with an array of other possible interpretations that aren't easily seen as possessive:
bildør - a door that is decorated with pictures of cars.
bildør - a door that is made of bits and pieces of car(s).
bildør - a door that leads to a car.
bildør - a door that is shaped like a car.
bildør - a door that is meant for cars.
These interpretations are not very common, but there is nothing inherent in the compound itself that prevents them. The meaning of compounds in general is quite malleable given the right context. That property is in stark contrast to constructions like 'Karis bil' or 'Kari sin bil', which are unequivocally possessive. So the question is: do you want to use the same label for constructions that carry possessive meaning
sometimes (compounds) as for constructions that are
always possessive (Karis bil, Kari sin bil)? The answer will depend on whether you focus on meaning in context or meaning without context. Your choice
As for the translational side of it, languages can be sensitive to various aspects of the possessor-possessee relation. Distinctions that are made in language A are perhaps not made in language B, so when you talk about possessives in Romance languages being
translated with exactly the same meaning into Scandinavian languages,
such differences are sometimes made invisible. French, of which you gave one exemple, largely uses the preposition
de to express possession, but does not necessarily encode information that a speaker of Norwegian would be sensitive to such as definiteness, animacy and alienability:
-
Karis bil --- I don't use this construction myself, but to the extent that I have intuitions about it, the 's' sounds very awkward (at least to me) with non-definite expressions: en venns bil

. It works better with definite expressions (naturally including names): vennens/Karis bil.
-
Kari sin bil --- This construction works well for animate possessors (humans and animals), and is rather awkward (or even ungrammatical) with non-animate objects: bilen sin dør

.
-
Bilen til Kari --- This construction works well for alienable possession, but inalienable possession is akward: døra til bilen

. If you want to express inalienable possession, it is better to use the preposition
på: døra på bilen.
The alienability property is perhaps clearer if you compare expressions like 'vekta på Kari' versus 'vekta til Kari'.
So in sum, translating with exactly the same meaning is not a trivial thing
