In English, I can say both 'year of birth' and 'birth year', so can I feel free to swap related words in French deleting 'de', for example, from 'année de naissance' to 'naissance an'?
You can't do that in French. Année de naissance is the only option.
The first point is that naissance is a noun, so obviously, it cannot be used as an adjective.
Secondly, the idiom to translate birth year is "année de naissance" so it is what you have to use as @vc74 says.
Finally, if you really want to have an adjective related to birth, even if it has not this meaning, it would be "native".
So the closest litteral translation of birth year could be "année native".