I think Daniel (pylot) and Jon (pitangadiego) nailed it on the head, but I'll toss in some personal opinion and experience.
I think a major part of the problem is name confusion. Specifically, The name Brown Turkey is so widely recognized that it ends up being improperly used all the time - in the produce section at the store, in the orchard/yard, and at the nursery.
Mutations, genetic similarities and climate they're grown in could also have some bearing on it, but I think those issues are dwarfed by name confusion.
For examples of those three places and how they cause problems...
At home, your average person (like, 99% of the world) doesn't know anything about figs. The name Brown Turkey has become interchangable with "fig" to them for a numer of reasons. After telling some people I grow figs at home, I've actually gotten the response, "Oh, you grown brown turkeys?!" ... they actually believe that every fig is a Brown Turkey. I've also bumped into home growers in Atlanta who actually have Celeste trees, but they call it a Brown Turkey because the fruit is brown (!!!) - after all, "it's brown skinned, so it MUST be a Brown Turkey", they tell me. Again - general ignorance. Big problem.
Several people have been shocked when I tell them that Brown Turkey figs are actually purple/green when ripe, then their jaw drops when I inform them there are well over a dozen different fig fruit colors and I'm growing several dozen varieties at home. They're floored when I tell them there are hundreds of varieties to grow. I honestly believe this is probably how we end up with awesome varieties like Brown Turkey MD - that is, someone has a fig that's not actually a Brown Turkey, but they call it a Brown Turkey, then cuttings and suckers are shared, and before you know it, everyone thinks Brown Turkey is awesome because of this one person's mistake. We end up with names like "English" Brown Turkey, "Improved" Brown Turkey, "California Brown Turkey", etc. which I personally doubt are actually a Brown Turkey, but who knows anymore?
At the produce section or farmers market, the customer's familiarity with the Brown Turkey name lends to fig fruit being incorrectly labeled as Brown Turkey to help sell the fruit. After all, names and familiarity sell - with names in particular, how many people have you met that buy a fig tree because it's labelled "Strawberry" or "Raspberry Latte"? The visual you get when you hear those names is pretty powerful.
I've seen our local stores AND farmers markets selling Celeste figs as Brown Turkey on a number of occasions. When I bring this mis-naming issue up with them, they nod and smile, but I'll come back the following week only to find they're STILL labeled Brown Turkey - they usually don't care. This pigheaded mentality undoubtedly leads to buyers loving the taste of "Brown Turkey" (after all, Celeste is a pretty good fig) and they run to the nursery to ask for their own "Brown Turkey" tree.
I think retail nurseries are most to blame. The familiarity of the name Brown Turkey means trees that actually aren't Brown Turkey end up being labeled and sold as Brown Turkey because it's the popular/familiar name customers ask for. Again, the name sells, and most nurseries are looking to sell fruit trees, not a reputation, so they don't understand or care about the importance of variety. I believe this is how we end up with some pretty awesome figs being labeled as Brown Turkey, but very unlikely to be an actual Brown Turkey. Stores like Home Depot and Lowes are often to blame for us in the southeast, but other major chains have been just as bad. Sometimes it's not their fault - their source mislabels - it's unfortunate.
When you consider this, is it any wonder some people say they've had good tasting Brown Turkey figs? Hell, they may have actually been eating a Black Mission, Celeste or Black Jack fig instead (noteworthy: Black Jack and Black Mission were found to be genetically similar to Brown Turkey by UCD some time ago).
It's worth noting that my only personal gauge on what Brown Turkey is would be the tree I have from Monrovia (a grower), which has many of the general documented characteristics of Brown Turkey, including the sweetness, wateriness, complete lack of richness, fruit color, interior void and color, fruit size, dominant leaf shapes, cold tolerance, growth habits, etc. It really is a piss-poor fig in the flavor department. There are dozens of better tasting figs out there with similar or better growth habits that it makes absolutely no sense to grow an actual Brown Turkey fig in my opinion.
Reality is, my Brown Turkey from Monrovia may not be a Brown Turkey at all, so I could be spreading misinformation - Gorgi brought this point up recently (i.e "how do we really know what is actually a Brown Turkey"?).
In my case, my Monrovia Brown Turkey it sure tastes worthless, though, regardless of whether you dehydrate it (it helps, but not enough), cook it or preserve it, so I've considered trashing it on a number of occasions. I'm just not sure if anything will grow well in the spot I've got it.