I am actually active in the field of astrophotography. What you're describing is a common-enough opinion, but it's inaccurate to the point of being misleading.

When imaging outside of the visible spectrum, you have no choice. You must translate it into visible light, or else you could not see it. So, yes, there are many false-color images out there, and for a good reason - they show us the invisible part of the EM spectrum.

Of those taken with visible light, there are several categories.

Some of them are indeed false color. Examples: imaging the Sun with a narrow monochromatic filter such as hydrogen alpha, or some "magic" wavelengths like 540 nm. These should be displayed in their respective colors (red for hydrogen alpha, green for 540 nm, etc), but often arbitrary color choices are made, because the original image is monochromatic anyway, so it should not matter what color is displayed in as long as its nature is understood by the viewer.

And then there's a very broad slice of the pie where the colors are basically real, they're just not perceived that way by the human eye, or even by the camera - and yet they do represent the original spectral content of the incoming EM radiation.

For example, when observing a galaxy or a nebula, all the human eye can see is a black-and-white smudge. Even in a telescope, you almost never see color with your own eyes when looking at these objects. Why is that? Because we have two kinds of light receptors in our eyes, and the ones that actually see color do not work in low light conditions - we are all colorblind there.

But attach a camera to a telescope, have the instrument track the nebula for a long time, open the shutter and let the sensor collect photons for a while. Then look at the result. Suddenly, you see colors. Why is that? Because with long exposure, the camera can actually see colors. That's something that our eyes cannot do.

But are those images "enhanced" somehow in post-processing in ways that make those colors unnatural? It depends. For objects like the Orion nebula, or the Horsehead, the colors are strong enough to not need much post-processing. In other cases, you may have to apply fairly strong contrast and saturation boosts.

So there's a whole continuum of images taken in visible light, where the amount of post-processing varies from quite lightweight to pretty heavy. With some experience, you can tell which is which. I've taken images of the Moon in high resolution and I've left it alone in post - you can tell because the colors are drab, it looks like the surface of an asphalt road. I've also taken moonshots where I've applied very heavy saturation boosting - you can tell because you can start seeing colors in an otherwise gray landscape. This allows you to detect surface, terrain, and mineral features that would not be apparent otherwise, things like ejecta from impacts and so on.

http://i.imgur.com/le5vrzh.jpg

And finally there are many, many images of planets such as Jupiter or Saturn, where the colors basically match the imagery taken by space probes hovering nearby, and would match what people would see with their own eyes if they were in actual orbit there.

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TLDR: The real answer is "it depends". Some colors are totally fake. Some are real, but boosted. Some are real, it's just that the human eye could not see them due to our scotopic vision characteristics. And others are actually quite realistic in every way. It's a broad continuum. You learn to tell them apart by being involved in actually making them.