I'm not sure there's any kind of rigorous study of how, exactly, that imagery became associated with nuclear power specifically. But if I had to guess, it would be because:

a) most people don't spend any time or attention thinking about non-nuclear power plants, so once they have an imagery associated with "nuclear power," it's not going to be unsettled just because some coal plants look similar or not, because they aren't thinking about coal plants in the same sorts of images;

b) nuclear plants, for economic reasons, tend to be very centralized and ideally quite large in terms of heat output compared to other kinds of power plants, so while a coal or natural gas plant might have a large cooling tower, they are often small enough not to need them, whereas a nuclear plant much more frequently requires several, large cooling towers that are very visible;

and, the most "historical" answer,

c) a lot of the imagery that people associate with nuclear power appears to have emerged out of the wake of the Three Mile Island accident in 1979. Three Mile Island was front-page news globally for weeks and its cooling towers featured prominently in much of the imagery around it. TMI looks damned-near identical to the Simpsons nuclear power plant, for example.

Once you have a recognizable "image" for a given technology, it's damned hard to shake it, because one of the goals of using imagery is to evoke a familiar and easily-recognizable association. So when the Simpsons wanted to show that Homer worked at a nuclear power plant, they could have used some very different plant design — like the Fukushima power plant, but it's not surprising they would have gone with a TMI-clone, since that is what most Americans by then associated with "nuclear power" and something modeled on a plant like Fukushima would have required a lot more visual explanation for it to immediately make sense.

Perhaps more straightforward would be to associate the containment dome with the plant (those are, in American models, usually squat cylinders with rounded tops), since those are uniquely associated with nuclear plants and are present in a variety of models. The exterior shots from The China Syndrome (1979), which came out days before the TMI accident, are of a plant that (to me) looks like it is meant to evoke Diablo Canyon, which because of its placement on a coast does not have big cooling towers. But in the pictures of TMI, it takes a little work to pick out the actual reactors, so shadowed are they by the cooling towers.

I suspect the Simpsons-TMI associations are probably not, er, accidental (sorry about the pun), given that the plant Homer works at is not exactly supposed to be a great example of safe nuclear energy (indeed, the opening sequence involves a spent nuclear fuel rod of some sort being bounced around town because of Homer's carelessness).

I'm focusing on the Simpsons not because I think it's necessarily the vector of this kind of image, but it's certain a vector, and a big one. I would also just note that if you were near a nuclear power plant, you tend to know it: those cooling towers let off a volume of steam that can be seen for many miles around. Yes, there are exceptionally large fossil fuel plants that can have similar cooling structures, but it takes a very big plant to require cooling structures of that size. My sense is that while some coal plants do use such towers, they tend to only have one such tower, as opposed to the two-per-reactor approach that is pretty standard in the US for nuclear power, if they have a cooling tower at all.

Anyway. What I am saying is, a) I don't think it's that incorrect an assumption (the incorrectness lies mainly in the fact that many nuclear power plants don't use such towers, not so much that if you see a bunch of those cooling towers, you'd be wrong to wonder if they were nuclear in nature), and b) that I suspect that association may have developed primarily as a response to the publicity that surrounded the Three Mile Island accident, and if TMI had used a different cooling mechanism, perhaps a different imagery would have developed. My suspicion is, in fact, testable: one could round up nuclear power imagery before and after TMI and see if there was a noticeable difference. It would be an interesting bit of research (not just for this question, but regarding nuclear imagery in general), but I'm not sure anyone has done it.

There is a very nice literature on nuclear imagery, but it is focused less on these kinds of graphical associations and more of the way in which imagery is used for thinking through the promise and threat of nuclear technology. For this, the starting point is Spencer Weart's Nuclear Fear: A History of Images, which I very much endorse as an in-depth book and a good read.