Although I appreciate it's easy to miss, this is working correctly. When the doubler is active, during a race rings are collected one-by-one as you have seen. However, it is only once you banked your rings at the end of each track, will the Rings be doubled.
Once the race ends and the rings are added to your overall total, it's at this point you should notice that your Rings have been doubled.
I hope this helps clarify exactly what you're seeing, and I hope you continue to enjoy the game.