The position shown there is the position of the pivot, which is, as you can see in the image, where the camera's gizmos are - in your case the "scale" gizmos which come out of the camera itself. not where your circle is.
The reason you are seeing these numbers is because of the camera's "size" parameter. This parameter (used for orthogonal cameras) says how many unity units will be from the middle of the screen to the top of the screen (and of course from the middle to the bottom). In your first image, the size is set to 5. Notice how there are five horizontal lines between the y = 0 horizontal line and the camera object. This is also why the camera's y position is 5. You aligned the bottom edge with y = 0, and the screen middle is 5 units above that, therefore it's at y = 5. The reason it's 5.01 and not 5 exactly, is because you didn't align it 100% correctly, the corner you circled is actually at (0, 0.01).
On your second image, the size parameter is set to 6. You aligned the bottom edge with the y = 0 horizontal line, meaning the screen's middle (and thus position) will be at y = 6.
The reason the first image doesn't translate to a clean integer is because of the aspect ratio. 16/9 = X/Y, here you know that the middle of the screen is (in the first case) 5 units above the y = 0 horizontal line (because of the size parameter). 16/9 = X/5 => solve for X => X = 8.88888...
In the second case though, you selected 6 as your camera's size, and so 4/3 = X/Y => 4/3 = X/6 => solve for X => X = 8. That's why it's a clean integer.
In Unity, by default, the constant axis for orthogonal cameras is the vertical axis. This means that if you change the aspect ratio, the width of your camera's view will change, but the height will remain constant. Knowing this, you can do what you want to. It will however require a small script. You can technically stick this on any GameObject, but you may as well stick it in a script on your camera.
float aspectRatio = Camera.main.aspect; //(width divided by height)
float camSize = Camera.main.orthographicSize; //The size value mentioned earlier
float correctPositionX = aspectRatio * camSize;
Camera.main.transform.position = new Vector3(correctPositionX, camSize, 0);