I'm glad to see that I'm not the only one giving some thought to riddles like this one.
Both the civilopedia entry and the manual (p. 98) regard Medicine as an advance of antiquity. Shakespeare being part of the Rennaissance, it's hard to see any direct historical link at all. My speculation is that the connection is an intellectual one (like the Pyramids not actually allowing concepts like communism and democracy to develop, but rather being the result of a highly efficient government). In a very abstract sense, both Hippocrates and Shakespeare are about the question: "What is man?" (The fact that Yoric's skull is used to represent the Theatre points to Hamlet.)
I wonder what answer Sid Meier or Bruce Shelley would give if they were asked something like that. There are tons of questions like this one that are never asked. I tried to think of a gameplay-related reason for the Theatre being attached to Medicine and not any other advance, but I could not think of one.