Does discretization of continuous features also lose information about distance?
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?