I don't know exactly what is the correct/technical name for this effect, but, to illustrate, i'm searching for a way to reproduce this effect in OBS for a live video source (both window, webcam and so on).
I don't believe OBS can live blurring effect. The effect you want is actually called compositing where there are two layers and one does not have a blur effect and the other one does.
Donkey Kong background is actually scaled and blurred.
This can be done in editing but not live.
Check out Blackmagic Design website and look for free editing/color correction software. It is called Davinci Resolve.