How to capture a high quality screeshot in 1920 x 1080?
The reason you're having problems is that you are currently using "Retina-mode". This means that the systems outputs a high resolution to the display, but makes it "look like" a lower resolution - which is what is stated on your screenshot. I.e. the display might "look like" 1280x800, but the actual resolution is 2560x1600.

The idea is that the system render text at a much higher resolution, but changes the font size to match, so that the output text is at the same physical size as if you had the low resolution, but it is displayed at the higher resolution meaning that it will be sharper, crisper and easier to read.

In order to change the real resolution to 1920x1080, you'll need to use SwitchResX and change the resolution to 1920x1080. However, take care that you choose the option that doesn't say "HiDPI" to the right. This is the name SwitchResX uses for Retina Mode. If you choose 1920x1080 at the bottom of the list where it is stated without HiDPI, you'll get that real solution - and screenshots are then taken with that resolution.

Doing it this way introduces no artificial blurring, stretching or anything of the sort. Text and everything will be rendedered in 1920x1080 without modification. However, the text will not be as "clear" as when you're using Retina-mode - but you cannot expect more with a 1920x1080 resolution.