Also, as a shotgun (sporting clays) instructor, I have found that many people do not have a truly dominate eye. This is more common in women. In these people, whichever eye is convenient is dominant.
For shotgun the reason the dominant eye is important is that your eye is the rear sight of the firearm. So if you are shooting right handed and are left eye dominant, then your rear sight is about 2.5" left of where it should be.
As I mentioned, my wife is left eye dominant, but can be situationally right eye dominant. This is easily demonstrated with an overhead incoming high clay target. As she tracks it, when the gun gets to a certain elevation, the muzzle jumps about 1" to the right as the right eye takes over. It is perfectly repeatable. The solution is a product called Target Dots, which work the same as the scotch tape over the center of the glasses. It removes her right eye from being able to see the muzzle and the target near the muzzle, but preserves her binocular vision for outside the muzzle area.
And while dominance is NOT specific to which eye sees better, in certain circumstance, that can affect things. When I shot clays a LOT, I starting having trouble with left to right crossing targets at eye level. My wife, looking over my shoulder, could see a muzzle jump like she had with the overhead incoming targets. So after some playing around, I decided my right eye had changed vision slightly. I went to my optometrist and told him my right eye had shifted 1/4 diopter.
He told me it was impossible for me to tell that. So he put me on the automated box and it in 6 checks, it said 3 times my vision was the same and 3 times, it had changed 1/2 diopter (the resolution of the machine). So he did the traditional exam and found......... my vision had changed 1/4 diopter. Got new right lens for my shooting glasses, and all was well.