Remember, though, that Drakes are still roughly humanoid in shape in their dracoform. Armor designed for humanoids tends to focus on the chest and back, because those are the largest parts of the body most open to getting shot. When you are dealing with any quadruped, though, the most vulnerable parts are on the sides, and the arrangement of limbs is different. And when you get into other animals, such as birds, serpents, or anything aquatic...
So while a Western Drake may be able to get their lined coat to work with their dracoform, a wolf shifter would find a lined coat cumbersome, at best.
Now, you CAN make armor for their shifted form (they have armor for K9 dogs, after all), but that's another kettle of fish.