Yes, now I get the problem. Not sure when it got dropped though...
But here is the GIF from the CS.
Thank you for digging this out! I completely forgot about this behavior.
But I must admit though (but adding a vote), that if gets implemented back, this might look as a bug for users who do click descriptors just to select a corresponding field, but expect nothing more.

Perhaps i was not clear the first time out... the clicking of either of the descriptors works as you show in your GIF but i am referring to when you the dialog comes up and you want to enter NEW dimensions, not those held from previous operations...it it here that clicking and entering the width and the transferring the newly entered value (by way of a simple click on the height descriptor) fails.
If you have used the older versions of Adobe Illustrator you will know this shortcut and time-saver

This is strange. I am on Windows and it DOES work for me.
See the GIF attached.

please bring back the one-click duplicate entry for dialogs. This was available from version 8 through cs5. Make it a toggle in preferences.
for example if i wanted to make a square 100mm x 100mm i would select the rectangle tool (M) and then click once in the work area...the dialogue box would come up...I would then only have to enter 100mm into the width
then i would click on the height descriptor (not the white entry portion) and my dimension would be automatically duplicated in the entry portion of the height dimension

OK, so NEITHER “Alt” (to make a copy) nor “Cmd” (to scale proportionally) works in the PROPERTIES > TRANSFORM fields. BUT Alt and Cmd work INDIVIDUALLY in the TRANSFORM floating window(?). How it should (and used to) be, is that they should all work COMBINED or INDIVIDUALLY across all TRANSFORM input fields (top bar (Essentials Classic), floating and properties panel). So that holding “Alt + Cmd” would scale proportionally AND make a copy/new instance of the object with the new PROPORTIONAL values (as if “link” icon checked). Just as it does when manually (arbitrary) dragging and scaling, holding “shift” + “Alt” ...

Another lost feature is that when entering a value numerically for any of the drawing tools (Ellipse, Rectangle, Polygon etc) clicking on the name of the next parameter would COPY that value to the second box. For example, If I wanted to make a 60mm wide square, I would select the tool (m) click where I wanted it and enter 60 in the ”Width:” cell, then just click on “Height:” and it would copy the other value. Now; nothing ...

Found AI9's 'cheat sheet'.
Shift+Enter: Apply value in field, keep field selected (amazing! and it doesn't work)
An yes, not Ctrl+Alt+Enter, but just Alt+Enter — AND IT WORKS :D

I think this Ctrl+Alt+Enter feature can be partially recovered with Autohotkey+Javascript, but native one would be cool (again then).

@Egor 100% sure, in Adobe Illustrator 9 and earlier, not exactly sure when it disappeared since I haven’t always upgraded to each and every latest version. Feels like the original coder team retired and this feature was lost in the CC rehaul :-/

Local units is cool, but I still lack other units to be local, such as type and stroke.
I have both mm-mm-mm opened with px-px-px, and only rulers stay local when I switch between those.

Are you sure Ctrl+Alt+Enter used to make a copy of an object right from the transform fields? Never heard of this, but I like this one. Voted.

BTW, thank you for making Units local (to each document), and not Global, a feature I requested a few years back.