External display resolution does not match what is specified in Mac OS settings

What you're describing is macOS' "Retina" HiDPI scaling feature. The actual resolution that the GPU renders, the display sees and what is communicated over the display-cable is 3840x2160.

In simple terms, the resolution reported to applications is instead 1920x1080. This means that the actual usable resolution you'll find as a user is 1920x1080, but for example text is rendered "crisper" by macOS taking advantage of the underlying full resolution. Similarly images and video material can be shown in the underlying full resolution.

If you open System Preferences and select Display, then you see the scaling option you have chosen for your monitor. You can hold down the option/alt key while pressing the "Scaled" radio button to reveal a longer list of possible screen resolutions. You want to select one marked "low resolution" (i.e. not a HiDPI one).

You can also use third party utilities such as SwitchResX to get even more options for screen resolutions.

You can also disable HiDPI scaling completely from Terminal.app by running the following command:

sudo defaults write /Library/Preferences/com.apple.windowserver DisplayResolutionEnabled -bool NO

You'll need to reboot afterwards.

To reverse and enable HiDPI again, change "NO" to "YES".