4k at 200% does actually not work as described in the question; to make it work like that you could simply operate the 4k screen with a FullHD resolution.
The 200% scaling works differently. Unfortunately I am not well versed enough in the inner workings of ubuntu's display management to be able to explain how exactly it works, but from my experimentations I can say that yes, it does indeed render most things in a hidpi mode, where fonts - and icons, in some cases - are rendered with a higher resolution. Practically all of the "standard", builtin ubuntu GUI, including preinstalled applications, are rendered at double size, at the given resolution. This includes the system settings, file browser, I can provide a few images and highlight the differences between 100% and 200% scaling, to make things clearer.
Closeup comparison of a launcher icon in the dock. Left is 100% scaling, right is 200% scaling. The left half has been scaled up x2 in gimp for comparison purposes.
Not much to add here. As you can see, the icon has visibly more detail with 200% scaling.
Closeup comparison of chrome and text body of the standard text editor. Left is 100% scaling, right is 200% scaling. The left half has been scaled up x2 in gimp for comparison purposes.
Icons, labels and text body are all rendered with visibly more detail.
Comparison of GIMP, a foss image manipulation program. Left is 100% scaling, right is 200% scaling.
Here it gets funny. Text elements are all rendered at twice the size, as expected. The icons however, are not, which might make them potentially too small to use on a very high dpi screen, despite the scaling. Notable is that the size of the image preview is not affected by the 200% scaling, which is certainly correct behaviour.
Closeup comparison of PyCharm, an IDE developed by JetBrains. Above is 100% scaling, below is 200% scaling, respectively. The above has been scaled up x2 in gimp for comparison purposes.
As the answer expresses interest in a JetBrains IDE specifically: PyCharm works very well with the 200% scaling. All user interface elements - icons, text elements, menu bars - are scaled up, rendered with visibly more detail.