How is it that "slim chance" and "fat chance" have the same meaning? Is there a linguistic classification for this phenomenon of seemingly opposite phrases that have the same meaning?
As John Lawler observes in a comment above, "slim chance" directly describes the chance of occurrence of some future event that is not at all likely. Slim falls into line with slight and little, both of which are frequently used with chance to express the idea that the odds that a particular thing will happen are not good. From the expression "slim chance," we also get the lengthier descriptive phrase: "two chances—slim and none," sometimes with the additional flourish "and slim just left the building."
"Fat chance," meanwhile, uses fat in the opposite sense to its logical meaning of "large." Christine Ammer, The Facts on File Dictionary of Clichés (2006) has this entry for fat chance:
fat chance Practically no chance at all. Although fat in this context means "good," the term is always used ironically to mean hardly any opportunity. A slangy Americanism of the twentieth century, it was used by P. G. Wodehouse in Laughing Gas : "A fat chance, of course. I should have known his psychology better."
In effect, "fat chance" and "slim chance" present the reverse case to auto-antonyms (words that can have opposite meanings, like cleave, oversight, and sanction); in this instance, the same meaning is attributed to seemingly opposite words or phrases.
Since there are so many category names for auto-antonyms—Wikipedia lists addad, antagonym, antilogy, contronym/contranym, enantiodrome, enantionymy, enantiosemy, Janus word, and self-antonym—you'd think that some classification term for the reverse case must exist. But I haven't found any that appears to be in widespread use.
The author of the Antagonyms webpage points out that the words cool and hot (as applied to something attractive) and the phrases "burned up" and "burned down" fit a similar pattern to "fat chance" and "slim chance." The same might be said of inflammable and flammable (both of which can mean capable of catching fire). The Antagonyms writer suggests that opposonyms or pseudopposites might be a good category name; but to judge from the few Google search matches that turn up, neither term appears to have gained much of a following in actual usage since 1999, when the page was last revised.