une culture culinaire qui lui est propre

In the lesson linked to in your post, it is explained that propre can also mean "own", depending on the position of the adjective.

Je dors dans mon propre lit. -- I sleep in my own bed.
Je dors dans mon lit propre. -- I sleep in my clean bed.

In the case at hand, propre is separate from the noun. In this case the meaning is determined by context. A "clean" culinary culture doesn't make much sense, so it's one's "own" culinary culture, or -- better yet -- one's specific culinary culture.