Searchlight itself was also among the production companies, according to some sources, although we'd probably have to study the end credits in some detail to figure out how true that is. Regardless, in the world of indie films, the distributor is the brand most associated with the movie and is the one responsible for promotion and negotiating screen counts and all that good stuff. Production companies, like Element Pictures in this case (who consistently make great stuff), are a little more inside baseball. You really have to be paying attention to know who produced what. The presence of TSG does indicate that Disney put some non-zero amount into the $35 million.
From my reading, distributors typically get a % of the gross, not of the profits. So in this case, the statement about Poor Things needing to make ~$87 million (2.5x budget) in order for Disney to see any profit is most likely incorrect, but it would depend on how much they paid Element, etc. for the Lanthimos deal that they have in place -- Searchlight did The Favourite, Poor Things, and the upcoming Kinds of Kindness -- and how much they spent on marketing. Hitting $87 million would only serve to absolutely ensure that a profit was made.
