1) Start the game
2) select Hot Lapping mode
3) go to Garage
4) change the car skin to any other
5) go back to main entry screen nad select Single Player mode

...in the top right corner you can't see the skin you've selected in Hot Lapping mode but if you go to garage, the skin is OK.

I tried that with 2) - single player and 5) - hotlapping and it worked fine.

I reproduced this, but while doing so, I found another issue.

I started the game and went to hotlapping mode. The track doesn't matter. The car that popped up was the XFG. I changed the skin, clicked ok, and clicked back to go to the main menu. From there I went to single player. The track was the same, but the car had changed to the XRG (Is this supposed to happen, maybe a dev can shed some light on it?). So ok, fine, I went back to hotlapping, and the car stayed the same, XRG again. So why is the car predetermined when I go to hotlap mode, then stay the same when I go to another mode, when the reverse scenario doesn't happen? This is issue #1.

Issue #2 is the one you mentioned. Since I was pretty much forced into the XRG, I went with it. I went to hotlapping and changed the skin of the XRG. Then I went to single player and the XRG which was revolving in the corner still had the old skin, but when I clicked 'Garage' it was repainted with the new skin. Nice find Flame.