I have been searching for matching algorithms similar to the Gale Shapley, or stable marriage algorithm, but with no luck. The question is if there are other algorithms that are similar to the GS and used for the assignation of persons or items based on preferences?
A very similar well studied problem is the assignment problem. In essence, it seeks to find a matching in a weighted bipartite graph that has maximum weight. Such matching is not necessarily stable, but maximum weighted matching might be preferable than a stable one, depending on the application.
On the plus side, the well known Hungarian algorithm solves the assignment problem in an optimal manner. There are also other methods to solve it such as the Auction algorithms. As I said, the field includes well established methods.
Maybe you can think about representing your problem as an assignment one rather than stable marriage.