I haven't seen this notion. Also note that Bachir did not use the whole space of continuous bounded functions for $\phi$ but certain subsets (doesn't the biconjugate get weird if you use the full space?).
Two pointers:
There are various notions of abstract convexity and some of them do feature a generalized notion of conjugation (I would start to look as Singer's "Abstract Convex Analysis").
There is the notion of $c$-convexity, and $c$-conjugation used in optimal transport (see the books "Optimal Transport for Applied Mathematicians" by Santambrogio or "Topics in Optimal Transportation" by Villani, for example). This does not generalize to vector valued functions, but may still be worth looking at.