Am I doing it wrong or do rented movies actually not support Family Sharing to another family member? Apple’s Family Sharing web page does not mention anything.
It should work but I think you will find that the rental period expires 48hours after the first device starts to play the rental movie. In any case there are reports of it not always working (note this was in 2018):
I’m afraid that example assumes the movie was rented on an AppleTV and the reply never actually says it works if you rent on one Mac and want to share to another. All the links there are to Apple pages that talk about sharing purchases, but rentals are to my knowledge never specifically mentioned. The movie was checked few hours after rental so the 48-hr limit was definitely not the issue. If I go check family purchases there’s simply no entry for this rental, hence no way to click on the cloud button to get it to play on another Mac.