I tested all colors and then set them to WHITE at 100%. Same deal. 5V all the way down, no drop at all. It was actually a little over 5 volts.
The two light strips are split at the controller. Both (I assume) receive 5 volts, which is 10 volts. A 20% headroom would be 12V, right? If so, they did it perfectly.
I think you have already all but proved that the controller must be using PWM.
The two strips would be connected in parallel - connecting two 12V strips in series would require a 24V power supply, and would go wrong if the two strips were not identical (voltage would not distribute equally across the two strips - your strips may be the same now, but say you decided one was too long, and cut some of it off).
If you were to connect those strips (all three colours, and without the controller) to a variable power supply (or, as you more-or-less suggested, to a fixed voltage power supply, in series with a variable resistor), and started with the voltage across the strip at 12V. The strip would be at maximum brightness, and with all three colours on, white. If you were to gradually reduce the voltage, then as you would expect, the brightness would reduce. So how bright would it be at 6V? And, would it still be white? There are some clues further back in this thread.
Without a variable power supply, cut one section (three LEDs) off of one of the strips and connect it to one of the 5V power supplies that you probably have lying around.
Another thing you could do is simply switch your meter to measure AC volts instead of DC, and repeat your measurement. While you are about it, try measuring the output voltage of the power supply by itself with the meter set to AC and DC, and see if you can understand what you find (clues this time in the thread I pointed you to before).