I think in both pool and shooting guns, where stance and eye dominance really matter - is when teaching someone totally new to the sport. Mostly to identify whether someone is cross-eye dominant or extremely strong eye dominant - since we generally/ideally want to orient our stance to our natural binocular (both eyes) center vision - and it can become a bit mechanically/physiologically difficult for these folks when they are first learning how to form a quality, consistent stance. Roughly 80% of the population don't have these issues and will naturally center up just fine, following basic conventional instructional methods. Personally, I think this is the very first step that any good instructor will help beginner students understand. If someone is cross eye dominant, they will almost always need some help to sort out an effective stance, otherwise they tend to end up with really awkward head/neck angles and fighting all sorts of issues. For shooting pistols effectively, it's almost impossible - in my experience these folks will have to either have to learn how to close their dominant eye, or learn to make their other eye dominant with both eyes open. Which is a nice segway to my next comment below.
One interesting aspect of eye dominance that comes up in both competitive shooting and pool, but which I never see discussed is - offhand shooting. As a right handed, right eye dominant shooter, when I shoot left handed (both guns and pool), its much more conducive to shoot with my left eye dominant, otherwise I'll end up fighting all the same issues that cross eye dominant shooters do. All can be tamed different ways, but my experience is that one can simply learn a bit about eye dominance and how to shift to let the same side be dominant and this will result in a much more natural stance, which is also much more consistent with how you stand/see the balls when shooting right handed. That might sound complex to do, but it's really not as difficult as it seems. It does take some work at first, like most new things - it may not feel comfy or natural at first.
Just some thoughts. Cheers