I was at the barn the other day when the parent of a student asked me if a horse that was shivering was "cold."
I asked her which horse and established that it was Cowboy, who is in his thirties, and he was indeed cold - his temperature regulation isn't what it used to be and old horses, just like old people, sometimes start to feel the cold more. (Don't worry - he got his nice warm blanket quickly).
So, yes, horses that are cold or wet do shiver, just the same as we do - but a healthy, young horse isn't going to get cold enough to shiver until the mercury really drops. They are more likely to feel the cold if they are also wet.
Shivering can also be a sign of pain or certain neurological conditions in horses - but it's easy to tell the difference. If they're just cold, then if you put a blanket on them, they will stop shivering fairly quickly. Cold horses may also stop shivering when they are fed.