Another possible set could be a preference sharing set, similar to an affiliated set, but restricted to sharing a small amount of data.

First-party sets seem to be appropriate when the user is more likely to complain about domains failing to share information than to complain about domains sharing that information.

One area where FPS could help sites act in a way that meets user expectations would be in making preferences act as expected across sites. If a user visits comicBookPublisher.example and sets some option such as "do not sell my personal information" they would be likely to expect the setting to also be in effect on the newMovieBasedOnAComicBookCharacter.example domain, because the two sites have the same well-known characters and other trademarks, and are seen as the same "party."

Some sites also allow the user to decline substance or gambling ads. A user would be likely to expect these preferences to take effect across a set of domains that they perceive as a single "party."

A preference sharing set would be allowed to share a small number of preference bits (4-5 bits?) across domains that are clearly understood as the same party by users. The amount of shared data would have to be small enough not to work as a unique identifier. Set operators would probably not be able to share all possible preferences, but could meet the expectations of most of their users who choose the most common ones.

(I don't see this as something that would be widely adopted by large numbers of co-owned domains on their own initiative -- more as a solution that site maintainers could apply in case users ask why their preferences didn't take effect where expected.)

cc @johannhof