It's not double blurring your game. Programmers can decide which part should be blurred in what strength selectively. Exactly because your eyes and brain blur in real world, the game tries to simulate the effects of blur too. In example in a racing game to get sense of speed, it can blur parts of the ground in example. If you play a racing game, your eyes do not blur the image. So its not blurred twice.

Overall its not the exact same, but it tries to simulate it like we try to simulate light in games as well.

When im looking at the screen and move my eyea across it, my eyes blur.

It does not blur the same way ingame blurs. First this is not motion blur of the subject, its blur because your eyes move. This is different. Don't forget you have a brain too, which blurs things differently than what you just see. For motion blur, it depends how fast it moves and how close is it to the viewer. When you look on a screen, then you look at a single surface without depth information for your eyes and brain. And as explained in the other comments the subject your eyes look at stays still. The subject does not have any motion.

The blur you get from little movement looking at a screen that does not move is completely different from ingame motion blur that selectively activates motion blur on the scenery.