That was WAY too much.
If you carefully read the instructions on the label, you'll see that the normal recommended amount of detergent is a FRACTION of the cup they include. So why include such a big cup? If you need more, you could just use two or three cups if necessary, of a smaller cup, and it would be as convenient.
Answer: money. They're in the business of making money, which they do by selling detergent. And how do they sell more detergent: convince you to use more. One easy way to do that, include a cup that is WAY TOO BIG, knowing that most consumers are lazy and will just scoop, toss it in and go on.
Will it damage the fabric, almost certainly not. Will it irritate your skin, almost certainly so. Unless you included a couple of extra rinses to get all that gunk out of your clothes, you'll probably wind up feeling it later.
I use approximately half of the recommended amount of detergent and my clothes come out looking perfect. My solution: GET A SMALLER SCOOP. When you open a new box of detergent, find that new, huge scoop and throw it away! Keep using the small, right-sized scoop and save a bunch of money.