There is actually a formula to calculate what a person's oxygen saturation will be at altitude. The clinic near me doesn't test with lower oxygen, but does the formula based on sats at ground level and figures whether a flyer will be fine or short on oxygen. If your local clinic is actually using 15% (and I have never heard of this being done, but I'm not saying they aren't) it may be to simulate the amount of oxygen available to the lungs at a lower pressure that will be encountered in the airplane cabin. The airplane cabin will still be at 21% of the mixture, but it may be the EQUIVALENT of 15% at sea level.
It's kind of a tricky concept. If you breathe air at sea level, you are getting enough oxygen to get along at the standard 21%. If you were to fly high enough that the air pressure was exactly half what it is on the ground, you may not be getting enough to be safe, yet the mix is still 21% O2. There is just less of it getting into the bloodstream because there are half of the former number of molecules beating away inside the lungs. A pilot breathing regular air is safe to fly up to 12,000 ft, and then regulations state that if he/she is at that altitude for 30 minutes or more, supplemental oxygen must be used, usually a cannula with 2 or 3 liters flow. If a window blows out of a jetliner at high altitude and the cabin pressure escapes, the air up there is still 21% O2 and 78% Nitrogen, but it is WAY too thin to breathe, so the yellow masks with 100% drop out of the ceiling.
I want to take my cf friend to Hawaii, but at 25 she is already dependent on oxygen all the time. I cannot take a tank on board for her to breathe from due to the airline security problems these days, and the cabin pressure at 8000ft equivalent will be too thin for her, so I am looking into a portable concentrator that is allowed on the jet until we land and can get some rented tanks. Part of my ongoing outside-the-hospital-setting education is turning out to be just how much planning it takes to do activities that I take for granted everyday with a cf companion...