Consider the ray model of light. Let's say an object such as a pencil is illuminated, and consider one point on that pencil. Since there could be many rays of light bouncing off the same point on the pencil, one ray could hit the left side of your eye and another ray from the same point could hit the right side of that same eye. Why don't you see that point in 2 different places then? Thanks!
I believe that the answer to this question involves multiple parts. I will try and hit all of them. Since you mentioned the ray model, I will assume you are relatively familiar with geometric optics.
First, we do see different images of the same object at times! Or rather, we see a blurry image rather than a sharp one. If you bring a pencil so close to your eyes that you cannot focus on it using your iris (more on that later), then you will see a blurry image. This happens for exactly the reason you mention. Rays from the same point of the object take different paths to your retina. If the path taken to your lens makes a large angle compared to the path that passes straight through the lens, then in general the rays will not recombine at a single point, but will have some spread. As the object moves closer to your eye, the angles increase, so the spread becomes larger and you see a blurry image.
Your iris is very important for creating sharp images. Generally, the distance between your lens and your retina is fixed. The distance between the object and your lens is not fixed, but we would like to be able to resolve detail for some range of object distances, rather than just a single one. So, the only parameter your body can control is the focal length. By constricting and relaxing, the iris changes the curvature of your lens. The change in curvature leads to a change in focal length. So, whatever object you are attempting to focus on, your iris constricts so that the object is beyond the focal length of your lens. This ensures that the rays will converge toward the retina and produce an image.
However, even with the object beyond the focal length you still get a blurry image if the rays make a large angle with the axis that is perpendicular to your eye/lens/retina (as discussed before) and this is one reason why you have a pupil. The pupil only allows rays that are approximately all parallel to each other to fall on your lens. It effectively acts as a collimator. So now, you have an object at or beyond the focal length and a set of approximately parallel rays that fall on your lens and are focused. This leads to a relatively sharp image on your retina (assuming that the object is far enough away that the pupil can do its job).
The final piece of the puzzle is your brain. Even though your iris, lens, and pupil do what they can to create a sharp image for your retina, it is still imperfect. There are still a number of aberrations in the image that falls on your retina. Your visual cortex and related support areas in the brain do all of the processing and reconstructing that leads to you perceiving a sharp image.
Ideally, all the rays coming from a point source intersect at the same point, to form an image. When that point is on your retina, the eye can "see" the object (eg: a specific point on the pencil). Note that to form a clear/distinct image all rays must intersect at a single point.
Similarly, when you want to estimate where an object lies, you trace all those rays (which hit your eyes) backwards, till they all intersect at some point. That point gives you the location of the object. Given that they all meet at the same point on the retina (since they form an image in your eye), just extending each one straight backwards would not make any light rays meet. So you won't be confused by imagining there are two copies of the object.
Instead of light rays hitting two sides of the same eye, consider light rays hitting two different eyes. Then they can (and do!) each estimate a slightly different direction for the position of the object, based on the different images seen by each eye. That difference comes in handy for depth perception.