Similar case here, never used a protector for my iPhone 6 and after five years of daily use I never got scratches on it. One week after receiving iPhone 11 I already had light scratches on the screen. I'm always super careful with my devices and the only thing touching the screen is my pockets with nothing else inside or my fingers. After noticing the first scratches I was being even more careful but more and more scratches kept appearing on the screen.
I contacted Apple, explained my situation and the answer was "it's cosmetic damage and we're not going to do anything about it". I was a couple of days over the 14-day return period but I asked to return the phone anyway because paying $800 for a device that scratches practically on its own is just unacceptable. I got transferred to 4-5 different people and explained my situation to all of them, firmly but politely demanding to return the phone. Eventually I was told they'll make an exception and process the return for a full refund. They asked me a lot of questions about my order, whether I picked it up or had it delivered etc. They even asked multiple times if the phone had scratches on it when I first got it, which I found very weird.
I decided to take my refund and buy another iPhone 11 and had my local Apple store install a protector on it immediately. Upon receiving the second phone I inspected it very carefully while unboxing it and noticed that the plastic film covering the screen had fingerprints on it and small dust particles under it, which clearly showed that the phone had been unboxed and then put back together. I peeled off the plastic and noticed a small scratch on the screen, straight out of the box. This was clearly a used/returned item that was then sold to me as "new". I ordered it directly through Apple's website.
I went to the local Apple store, explained them that the phone has a scratch straight out of the box and asked to exchange it to a phone that's actually new. The lady said "no problem", processed the exchange and installed a screen protector on my new (third) iPhone 11. I clearly explained the reason for exchanging the phone and when her device was asking for a reason for the return, instead of "damaged on arrival" she marked it as "buyer's remorse". I'm sure that phone that's now had two previous owners will go back on the shelf and get sold to someone as "new".
For everyone who's getting these scratches, don't just post on forums. Contact Apple directly through their online chat or phone. Don't take no for an answer. They're aware of the problem but they're trying to shrug it off saying it's your fault. They have clearly released a product with a faulty screen that can not withstand normal everyday use. As more and more people get their new phones this problem is just going to keep growing until Apple launches a recall/repair program.