The phrase "milking it" seems to have originated in the context of finance. According to the OED, "milking" can refer to
The manipulation of funds for (esp. unscrupulous or illicit) financial gain; (more generally) exploitation of resources.
Similarly, given what the physical act of milking some animal is like, it makes sense to think about "milking it" as "getting as much as possible out of a resource."
All of this makes sense. I'm wondering, however, when this idiom started to be used in different contexts. When did it become common to hear "You're really milking that for all its worth" in reference to an injury, for example, or for an emotional slight?
Similarly, which is preferable: "milking it for all its worth" or "milking it for all it's worth"? Should there be a standard, or does it depend on the meaning one wants to convey? I'm inclined to prefer its, as a thing does possess a finite worth (theoretically).