If you think of the two parallel glass sides as canceling each other out you are pretty close to it. The first impact (low to high indices) does in fact disperse frequencies if the light is coming in at an angle, but the exits (high to low) mostly cancels that effect. There actually can be some small residual effects leading to small colored fringes.
The prism works better because the oppositely angled sides enhance rather than cancel the dispersion effects.
Here are the results of some further analysis and experimentation.
Part of the answer for why color effects are so hard to find when light passes through flat glass plates appears to be in the eye of the beholder... literally!
Here's the scoop: Angled light entering a flat plate should at first fan out its angles by color while within the plate. By symmetry, however, those slightly fanned out rays of colored light return to their original paths when they reach the second surface an re-emerge. So, the new rays will show essentially no difference in direction from their paths in the original beam, but will no be spaced very slightly apart from each other in rainbow order. For a typical plate of glass this separation would seldom be more than a millimeter, and for most glasses would be a lot less than that.
Now picture a point of light on one side of the glass and a human eye on the other side. Arrange both so that the line between then is at a sharp angle to the surface of the glass. Let's look at the ray going from the point to the center of you pupil. Your eye focuses that parallel white light as a single point on the retina, as expected. But When the angled glass plate is inserted, the same ray of light gets spread out along a tiny distance, usually much less than a millimeter. However, each colored ray in this bundle remains parallel to the original path.
This little sub-millimeter bundle then enters the pupil of the eye, carrying pretty much the same light as before, all traveling in parallel. What does your eye do with it? It forms the same white colored point image as before, since the light is all traveling in parallel. Think for example of red and blue light entering opposite sides of a magnifying glass: Both will end up near the center. There will be some chromatic aberration, sure, but it turns out that vertebrate eyes are very, very good at eliminating that form of chromatic aberration at the image level.
The bottom line becomes this: As long as the plate is not too thick, the physical separation of chromatic components will fall within the size of the human eye pupil, and the image will appear to be white - color free and pretty much just like the original, with just a bit more blurring.
That also leads to an experimental prediction that I have not yet tried: If you hold a pinhole in front of your eye while observing a pinhole light on the other side of an angled piece of glass, you may be able to see a short colored line instead of a white dot. I can't guarantee it, but it's likely enough that it would be interesting to try.
Now to the final part of the analysis: What if the glass is so hugely thick that there is no way the separated components can be captured by the human eye all at once?
Shouldn't that lead to some visible color effects, such as blue and red fringes on either side of a point of white light? Specifically, for a white dot or light, a blue fringe should appear towards the side angled away from the viewer, and a red fringe on the side of the glass that is closer to the viewer. For a black line on a light background this would be reversed, with the red on the glass-is-farther edge of the black line (since that is the nearer edge of the lighter part), and blue on the glass-is-nearer edge of the black line. (You can work out why that is with simple dispersion diagrams.)
But since the space-form chromatic separation effect is going to be small even for a quite thick piece of glass, where can you find something thick enough to show such fringes?
Fish lovers have conveniently provided a solution: They are called aquariums! The combination of glass and water makes a quite good approximation of a very thick piece of glass with decent chromatic dispersion.
But does it really work?
If like me you don't currently have an aquarium, here is a convenient online image of a see-through aquarium that is angled sharply away on the right. On the other side of the tank are both vertical bright lights from curtain folds (near the right side), and vertical dark lines from a picture frame (on the left). If you magnify the image, you will see blue fringes on the right sides of the curtain folds. Neither effect is intense, but both are definitely in this image.
If you do happen to have an aquarium, you should of course try it for yourself, since good direct experiment always trumps theory if they disagree! Don't look directly into a light, since modern LED lights are very bright and should never be gazed at directly. Instead, place a thin vertical stripe of white paper on a black background and illuminate that with a bright light pointing away from the observer. You can also try holding a small pinhole in aluminum foil in front of you eye to enhance any color fringing effect you may see.
And with that... I think I'll give this one a rest. Further discussion, especially actual results from experimenters with real aquariums, would be great though!