Not all fluorocarbons are created equal. The more manageable and limp the line, the more likely it will be stretchy. Invizx for example, very stretchy, easy to cast. Other lines like Shooter offer great abrasion resistance, little to no stretch, but it's difficult to handle because it is so stiff. It's not necessarily a myth, but it's not really true that fluoro has "no" stretch.
Stretching fluoro can indeed damage the line if you go too far. It will not bounce back to its original shape like mono lines will. Even with mono though, once you get past a certain point, it will weaken the line.