Most of the nebulae and galaxy photos are what we’d call false colour, yes - although it’s probably much more fair to the people who make these images to call them “exaggerated colour”, or perhaps “reconstructed colour”. These images do not usually reflect what we would see if we looked at them ourselves.
The human eye has a really bizarre sensitivity pattern to light. We’re pretty good at seeing things in the yellow-green range, orange we can usually do, but once you get into reds and blues, our eye suddenly gets really bad at registering the deep reds and dark purples, and our brain translates those colours into “black”, or more accurately as, “there are no photons here that I can deal with”. To anything outside the range of visible light we are completely blind. This odd sensitivity pattern means that it’s really hard to make a camera with exactly the same sensitivity as our eye. This is the same reason why it’s sometimes hard to get your camera to pick up the colours we can see by eye. Most cameras have settings nowadays to help change the sensitivity towards a specific colour, but they won’t perfectly replicate the eyeball.
Furthermore, from a scientific standpoint, replicating the eyeball isn’t an incredibly useful thing to do. We’re usually more interested in either a specific colour of light (usually one that corresponds to the colour certain atoms release) which helps us tell how much of that atom is present in the nebula or galaxy. Alternately, we can go after broader “colour” - the relative contribution of blue versus red light tells us things about the stars and dust in a galaxy. If we’re interested in these total values, and in trying to compare red and blue, then introducing the handicaps of the human eye into the equation will only serve to complicate our situation more than necessary.
Given that we’re detecting light at much better sensitivities than the human eye, and that we’re usually doing it discrete chunks instead of one (very complex) curve as the eye does, putting these chunks of light back into a single image is tricky business. Even when all the light was taken from the narrow range of light that we could see, it must still be reconstructed and tweaked to reflect the brilliance of the nebula in the colours we’ve observed. Hubble has produced many beautiful images (such as the one above) labeled as ‘visible light images’. What this means is that the narrow ranges of colours that Hubble observed all fall within the the visible range - but they have still been patched together, the colour of each set of data overlaying on top of each other to build an image in full colour. This particular image had 6 colours to work with, and it’s made a lovely and vivid image, but it is still only six colours. The colours here aren’t really “false”, but they have been “reconstructed” from six black and white images.
“Exaggerated colour” images can be used to extend our sight much beyond what we can actually see. Perhaps a galaxy is rather unimpressive in visible light, but has an impressive brilliance in the ultraviolet or X-Ray - to our eyes this is dark; but if the telescope can look at ultraviolet or X-Ray light, we can put it into our image, and reconstruct an image that we will never see with our own eyes.