Posting here just to document my issue along with everyone else.
I purchased the iPhone 11 at the Apple Store at the end of October, the day after it was released here in South Korea.
After just 2 weeks of use I noticed there were significant small scratches all over the display and remember simply thinking "it seems like this screen scratches a lot easier than my previous iPhones, this sucks..." but I am not that picky about minor scratches and they were not affecting my view of the screen so I let it slide.
Fast forward a couple more weeks, I start to notice more significant and slightly deeper scratching around the edges of the screen. I'm starting to think this is really strange because I treat this phone exactly the same as I treated my 7, down to wearing the same clothes so it was in the same pockets, etc.
Now, yesterday, I take my phone out of my pocket and am shocked as I see two huge, deep scratches running smack dab down the middle of the screen. They make a rainbow pattern and therefore affect my view of the screen. How they got there I honestly cannot explain. I always keep the phone in a pocket on its own and even if there was some sand/debris in my pocket there is no way it could have made that deep of a scratch in that location of the screen.
I always use cases on my iPhones but have never once used a screen protector since I have always found iPhone screens to be especially robust. My iPhone 7 never had a protector during the 4 years I used it and it looks better than my 11 currently does. Also, I work at a video game company and I have countless test phones I use and abuse constantly: right now I have a Galaxy Note 8, an iPhone XS, and a Galaxy 7... all of them, despite having been used heavily and tossed into my desk drawer constantly, are not scratched.
I even googled this in Korean and saw that this issue has also been documented in the South Korean media.
I'm going to visit the Apple Store on the weekend and see if they are willing to do anything about my screen. I'll update if I have any success...