Tl;Dr: The success of KY basketball is tied to their identity as a people; to steal the limelight, to out-success them, is to threaten their identity.
Like a couple others on this board, I'm from KY (but don't currently live there, however). Without poking any fun, even though some of this may sound harsh, this is the actual reality behind their obsession as I see it. This will probably be long.
Kentucky isn't just obsessed with Duke; they don't just hate Duke. They hate Louisville (local rival so this is understandable to a degree - more on that later), they hate Kansas, they hate UNC, they hate... see a theme here? Hardcore UK fans hate any program that consistently wins and the media spends anytime doting over in place of themselves. There are lots of facets to this and there are trends amongst demographics and their fandom.
As probably everyone knows, the vast majority of Kentucky is rural. You have Louisville and Lexington which are larger cities (particularly Lville), a couple cities the size of Bowling Green (home of WKU) with ~60,000 people, and then the rest of the entire state is nowhere close to that. All that to say, KY is very rural, which means that there is not much to do for entertainment and not much in terms of higher education, which means less opportunity in terms of basically anything.
Most (not all) of your Rupp Rafters/Cats Illustrated, hardcore-hate-everyone-else Kentucky fans are low-income, rural residents. They love UK basketball, their dad did, their dad's dad did, so on and so on. Hating the success of other teams -particularly Lville and Duke - is a cultural construct that partly defines what it means to be a Kentuckian; it's an identity marker. You're not just a Kentucky fan, you're also a Duke/Lville hater. They're not really separable as being a true KY fan means you hate those other two teams. For many of these types of fans, the obsession for UK basketball stems from both generational indoctrination and simply the fact that there is little to nothing else to look to both in terms of entertainment as well as something to place your pride in. There are no pro-teams of any kind to cheer for, and outside Louisville (and arguably Lexington), there aren't really any other kind of major attractions to distract you from being preoccupied with KY basketball. They're not "busy" with anything else in terms of entertainment.
For many of these rural hardcore UK fans, they have little to own and they have little to place their pride in. The mindset many have is, "We may not have much/may not be known for much, but dangit, we OWN basketball." Kentucky basketball is a part of their identity. Ask these types of UK fans what else they do for fun besides watch UK basketball and you'd be hard pressed to have anyone say anything besides, perhaps, hunting and fishing; because, for most, there just really isn't anything else. And again, for these rural hardcore fans, they spend all day everyday working, farming, etc. and so the big game on Saturday is *the* thing they look forward to that gets them to the end of the week.
When you're talking about middle-income people, in my experience, they're far less rabid of fans. Sure they're crazy about UK basketball and they still hate Duke and Lville, but they're far more tolerant and far more likely to joke around and take things more lightly. Of all my friends that are college basketball fans, all are UK fans and we're always ribbing one another and having a good time with it. They still want Duke to lose every game, but they don't want Duke basketball to simply not exist anymore.
That is all about fans outside of Louisville.
No surprise here, but most Louisville fans live in or are from Louisville with the rest of the state predominantly being UK fans. This is, of course, not true for all UK fans throughout the state, but there is certainly a racial/cultural element at work here in regards to the hatred for Louisville. Most of your low-income, very rural UK fans were brought up with the stereotypes typically characteristic with low-income, poorly educated, rural white people. Most Louisville fans are African-American. Most Louisville fans that are white are typically looked at as "thuggish" or "kinda rough". You won't really find a rural, "good-ol-boy" Louisville fan. You don't find too many African-Americans in KY that are KY fans. Obviously I'm speaking in generalities.
I think the racial element pairs with jealousy in their hatred for Duke. Many of those lower-income, rural, diehard UK fans dislike Louisville and Louisville fans "because they're all thugs" (and we all can read between the lines when that's said), and they partly hate Duke because Duke has a rep (as unfair as it may be) for being predominantly white and wildly successful. "If Duke is winning like that with 'all those white boys' then why can't we at the same rate? Why didn't those players come here, anyway? F*** Duke."
Again, middle-income fans generally have a different mindset here as well (though the Louisville/Louisville fan stereotype definitely transcends class throughout the state, imo). Their hatred for Duke is largely all in good rivalry fun in my experience. They hate Duke because Duke is a wildly successful program, K has been around a million years it seems, etc.
