Why does data visualization mostly use the Cartesian plane instead of the polar plane? Suppose there's a dataset that has features like the weight and height of people.
If we plot the dataset, we commonly use the Cartesian plane, the x-axis to represent weight and the y-axis to represent height; hence, points represent every piece of data.
From the case above, can we represent radius as weight and angle as height in polar plane?
It seems possible to represent weight as a radius, but representing angle as height seems weird since height and angle are different units.
That's why angle units may be radians while height units may be centimeters. So, is it possible to convert the height unit into an angle unit like a radian in this case through normalization?
If this is impossible, then what is an example of a dataset that uses a polar plane for data visualization?

Starting specifically with your example of height and weight:

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*The example doesn't convince on the simple grounds that angle is a circular scale but height is not, as you realise. You can ignore that easily by ensuring that your representation of height covers only some fraction of a circle, but even so any choice of a fraction is arbitrary and likely to prove awkward or lead to puzzlement or confusion. In particular, choosing a large fraction runs the risk of being misleading and making patterns hard to interpret, as data points for very short people and very tall people would be close on the graph.


*It is hard to compare densities on a polar scatter plot because data points are necessarily more crowded nearer the origin, other variations aside.


*Radius zero for zero weight might be an awkward choice, but any other choice would be arbitrary and quite possibly very awkward. That is, if the centre of a polar display represents (zero weight, zero height) that poses problems of some kind.


*The same kinds of comments would apply to swapping height and weight.


*Other way round, and more generally, you need a good reason for thinking that a polar representation is more natural, i.e. corresponding to the underlying phenomena and/or more effective psychologically.
Backing up, and coming at the question more generally, there are many interesting issues here of both principle and practice. There is an over-arching theme: it doesn't follow that the nature of the data space determines what is going to work well psychologically. In Tukey's terms we need graphs that are easy to understand in principle and effective in reading in practice. (A pie chart is a classic case of easy but not effective: small children get taught pie charts, and should be able to explain them, but that doesn't make them more effective than alternatives, as pie charts can make it remarkably difficult to discern even some simple contrasts, such as one slice being a little bigger than another. Pie charts as a polar display are especially germane to the discussion.)
Coming from another direction (+), there are many spaces with one variable defined on the circle or some part of the circle. Sometimes that variable is an outcome (which direction does an animal move? which direction does the wind come from?) and sometimes it is a predictor (how does direction faced influence temperature? how thick is the bark on different sides of a tree?).
Leading examples of circular scales are from the
compass: compass bearing or map direction
clock: time of day
calendar: time of year
The occurrence of c words here I find a curiously congenial and compelling coincidence.
But, but, but: even when the data space is clearly circular, at least in part, it can still be true that linear representations are customary and indeed widely considered easier to think about. So, we are all familiar with clocks and watches showing 12 hour scales (24 hour scales being unusual, but hold that digression), but plotting a time series in terms of time of day on a horizontal axis still seems much more common than plotting it as a trace wrapped around a circle.
Small and perhaps obvious tricks are

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*cutting the scale where it is least painful: if your concern is with phenomena that peak at night, don't start the scale at midnight


*repeating some values at the ends of the scale (lest someone posts this as a new trick, it goes back to J.H. Lambert at least).
The same holds for time of year. Florence Nightingale famously plotted various Crimean war deaths on polar diagrams (by the way, consciously repeating technique from her friend William Farr and accidentally repeating a design used by several others previously), but her data are just as validly plotted on a linear scale, despite numerous articles praising those graphs fulsomely on various grounds. (One of several considerations: The complicated patterns in her data reflect the irregular course of the war as well as seasonal variations (winter makes things worse for everyone) and changes over the period in medicine, hygiene, and so forth. Another: She showed variations for two years with two polar diagrams, which raises the need to compare between and within years, the same issue as with repeated pie charts.)
I have seen recently many spiral representations of (e.g.) pandemic infections, hospitalizations, or deaths. They seem to have attracted attention for the wrong reasons as being engagingly weird rather than because they are effective(++).
The same holds true for phenomena involving map direction. Linear representations can be easier to follow, especially if thought is given to where to cut the scale. (Sometimes North should be in the middle of a linear direction axis, and so forth.)
Here is an easy example where polar representations seem natural. Ornithologists attach identifiers to birds' legs and release them at a recorded origin. Some of those birds are later found dead or caught by other ornithologists, or even the same people. Or birds (or bears, or cats, ...) are tagged in a way that their movements can be followed remotely. Movements thus define a series of vectors with distance and direction. But even a polar representation is not necessarily best. Interested in the distribution of distances travelled? Reduction to a more standard graph is likely to be as or more helpful than eyeballing a set of vectors. Interested in how far mean distance may vary with direction?  That could be looked at using  a mix of sinusoids or a kernel smoother that wraps correctly around the circle, but again a Cartesian representation may help more, with where to cut the circle before unravelling the point of art. (Ever tried to gauge by eye whether two vectors on opposite sides of a circle are about equal or different?)
In modern biological informatics, tree-like structures are often stretched around a circle. I have less experience of those. It seems that readers quickly get used to that kind of graphic if they produce and see many examples. As often happens, what is  regarded as intuitive and what is regarded as familiar can be hard to tell apart. I reserve comment on several other bizarre designs.
(+) Pun intended.
(++) A while back a simple meme in some visualization circles(+++) was that graphics can provoke reactions of Aha! (good: the reader sees a pattern or helpful detail) or Wow! (good for the designer whose software skill or wacky originality is being admired). I always wanted to add Huh? (namely, what am I supposed to see here? this is just data art, or a hairball showing me this network is confused or tangled, or whatever).
(+++) Pun not intended.
