The default display resolution is 1920x1080. Switching to 3840x2160
makes everything tiny and reduces the refresh rate to 30Hz.
When you say "the default resolution is 1920 x 1080" - I presume that means the SCALED resolution. (Otherwise it would only use half the screen.) A screenshot of your System Settings would confirm this.
Here's my 5K display, which is listed as 2560 x 1440. But it's using all the 5K pixels, no matter which size I choose.
Your Mac is making everything twice the size by pretending that it's a 1920 x 1080 display, to avoid the tiny-ness of the actual native resolution.
In short: that's what you want, and everything is working correctly. You'll not have more real estate than than your old 1080p display, but everything will be twice as sharp. If you want a bit more space, then try an intermediary resolution, but things will be correspondingly smaller.