That's normal. The more driving skill you get, the harder it is to get it up even higher.
Try taking a lengthy custom race and you'll see an increase, if not then there might be an issue.
Winning or not doesn't matter on driving skill, only the track and the laps done does.
On ever higher driving skill it could take a whole day to get just 0.01 more. Nothing wrong with that.
Each race gets you some exp, but it might be a small increase (like 0.002) that won't be enough to increase your current 6.1 skill.
closing it for now, if there is an issue and no increase in a long time, mail me and we'll look into it.