I think of a shakeout run as any run that is slow enough to provides almost zero aerobic stimulus (though maybe you can pick it up for just a couple of minutes at the end), and will absolutely not require any recovery time, i.e., you should feel better a few hours after a shakeout than if you didn't do it
I think of the purpose as mostly biomechanical (loosen up the legs), but also somewhat psychological and even gastrointestinal. Short, easy runs also can stimulate a slight bump in growth hormone and, for reasons not understood, can help to lower your resting blood lactate level if it's still chronically elevated in the day or so after a difficult effort (which does happen and is entirely different from the acute increase in blood lactate from hard efforts, which is very quickly reversed after you catch your breath).
As others have noted, there are a number of places where runs like these can fit into your schedule. Immediately post travel is a great time for it, because sitting for a long period can shorten muscles, and sitting on a plane can leave your legs like giant water balloons. Early on race day is good, especially if you're used to morning running and your race isn't till much later. I also like to do 3 miles the evening after a long run or a race. I've found that any effort that's hard and long enough to do real muscle damage is going to leave you much more stiff if you don't get in your active recovery. My theory is that if you don't keep moving, you allow muscles to heal in shortened positions.
I agree that a shakeout is normally not just your easier run of a double, but it depends. When athletes are just starting to double, I usually insist that their secondary run be very easy and short at first. Also, some higher mileage marathoners will make their second run more of a shakeout because they try to more heavily load the other one, whereas middle distance runners might be more likely to do 6/6, both at a decent clip. And although I'm not sure what terminology Kenyans would use, I think that for Kenyans who do the classic three-run schedule, the pre-breakfast run is definitely what we'd call a shakeout