FIS governs the whole ball of wax with respect to the sport. So, One of their committees determines the rules for exactly what a FIS sanctioned SG race course, homologated hill, and course "set" will look like. They also determine the safety requirements, such as the netting. The rules determine how long a course should be, how much vertical drop, how many changes in direction {turns}, etc. There is some leeway for course setters, which is based on the level of completion. A "FIS entry" SG, for younger athletes, for example is going to be nothing like a NorAM, let alone a Europa Cup or World Cup.
At the same time, FIS regulations determine what the skis need to be for this skiing. Dimensions like the width of the waist, the minimum radius, minimum length.
The whole "thing" hangs together. To safely race, and to be competitive in these races and course sets, you need the right skis. You might think that "hey, I'd rather have a shorter ski with a smaller radius", and the problem is that it will "hook up" and want to make turns that are much tighter than the course set dictates. And "hooking up" at 50-60mph or faster is not good. It's a great way to get badly injured. Skis with more "shape" to them, and more sidecut, as many of us know from experience do not like to run straight too much. At all. They get sketchy, and are much more comfortable on edge. It's best illustrated by a SL ski, but a 30-35M GS ski is no bargain in a true FIS set.
You're not running straight for long distances in a SG, or very, very rarely. So the skis are not exactly like DH skis, nor like GS skis. I'd say they are more like DH skis, though. And in some SG sets, there can be some pretty open and straight sections, running in a tuck, etc. The skis matter a great deal in SG once you're racing at a fairly high level. For example at the top levels of U16's, they are critical. They need to run fast, absorb terrain and be responsive to the long SG turns.
I am perplexed when folks talk about how they love to free ski on SG skis. I have a big aversion to that. unless it's under the perfect conditions. wide, empty hill, perfect surface, and skilled pilots. Have seen some bad, sad outcomes.