Does discretization of continuous features also lose information about distance? During discretization, it "squashes" nearby values into one bin, losing a little bit of information along the way.

But doesn't it also lose information about distances of features? For example, if we have height as continuous feature, we can e. g. create bins very small, small, medium, large and very large. Isn't it problem, that once we have these categories, we lose information that very small and small are closer than small and very large?

This would be even worse for predicted feature - I would assume that regression that tries to predict height number would be much more successfull than classifier that tries to predict category of height discretized, because cost function of regression can account for a distance from correct height, but cost function of classifier could only answer "correct" or "not correct".

Yet I haven't found any mention about this when I've searched about discretization.

Are my assumptions incorrect?
You are absolutely correct. Also, sometime the binning could be arbitrary as well, but sometimes it depends on how you use the insight.

Using marketing as an example, we often see age grouped into buckets like 18-24, 25-34, etc.

Would a 34 year and 35-year old behave dramatically differently? Probably not. But then, it might still make sense to use such grouping, because many ad platform targeting tools use the same bin definition. It doesn't make sense logically but practically.

Of course, it really depends on the situation and the kind of analyses you need to do. For some distribution, you might want to do a log scale before you bin, etc.

Back to your example, if you feel that height should be a continuous feature and most models can handle it, is there a particular reason why you would want to discretize it?