On macOS you have both choices when connecting such a 4k monitor:
You can choose to have it run at a real 1080p resolution, where the signal sent to the monitor is really 1080p. In essence the image displayed on the 4k monitor would then be visually the same as had you bought a 1080p monitor of the same physical size.
You can also choose (which is most probably what you want) to have it run scaled to "Looks like 1080p". This means that the signal sent to the monitor is really a 4k signal, but user interface elements are scaled to be the same physical size as if you had been running 1080p. This ensures that the system is usable without requiring extraordinarily good eye sight. Note that this means that text is really drawn at the large resolution, which ensures crisp and sharp text, which is easier to read. Images and video are also displayed at a larger resolution (if available), which means you get better picture quality on the 4k monitor compared to a 1080p monitor.
In addition to the two scenarios outlined in your question and above, you also have the third option of running the monitor in a 4k non-scaled resolution. This means that you'll have a very large amount of screen real estate (i.e. room for lots of windows, content, etc.) - but everything will be 4 times smaller physical size - so this is usually only a good choice if you buy a really large monitor (like for example when using a 50 inch TV as a monitor).