If you go to the Amazon listing for Doom PS4 and
click on the link to see all the offers, then filter for NEW condition and PRIME shipping, you will see the lowest offer is Amazon for $19.99 but only available to Prime subscribers. Next lowest offer is $37.50 from Games 4 Life. Click on Games 4 Life and it takes you to his feedback page where it lists his location as "LYTHAM ST.ANNES, LANCASHIRE" which is in the UK.
So let's say a Prime subscriber orders the game from Amazon and they are out of stock in the nearest warehouse, but Games 4 Life has some in stock at that warehouse and they are the next lowest price. Amazon buys it from Games 4 Life and ships it to the customer so that they receive it within the 2-day shipping window or whatever. If the game was the normal US version there would be no problem.
The seller of this item printed out a label with the ASIN (Amazon number in the format B0xxxxxxx) B00M3D8IYM. Each page on Amazon has one of those numbers assigned and it shows up in the URL, you can also input that number into the search bar and it will bring up Doom PS4. Long story short, everything sold under that ASIN is supposed to match the details on that page exactly. Whoever stuck that label on the game is representing the product as the US version when it is actually the UK version. Now we don't know for sure that it came from Games 4 Life, but let's take a look at some of his feedback:
"Game arrived intact and works, but the version of the game is actually EU and not US, PEGI rating on the front versus ESRB. Luckily the Xbox One isn't region locked like the Xbox 360 was but without being warned about this, I was tempted to ship it back immediately without trying to play the game first."
By annybull on August 24, 2016.
"Item arrived late and Playstation bonus codes included in the case were rejected by Playstation Store so bonus items could not be redeemed."
By Mom on August 19, 2016
"Ordered Gears of War judgement for Xbox 360 and received a PAL copy. Luckily, this specific game, the PAL copy works on North American Xbox 360s but still not cool. The game was listed as a NTSC copy. It works so I won't return it but people should know."
By David D. on August 8, 2016.
"Received the European version of the game (PEGI 12 in the bottom left) instead of the advertised North American version (T for Teen in the bottom left). I was able to return with a full refund."
By VGKPfan on August 5, 2016.
"Item displayed was the US version and I received the European version which was useless to me."
By David Marcinkevich on August 5, 2016.
"These guys continue to ship non US videogames when the product clearly indicates US product. I keep returning. Avoid them."
By Molly on August 4, 2016
"I bought the game from Amazon US and the image is showing NTSC version. I don't know why I got a PAL version that doesn't work on my XBOX 360"
By Yousif A. on August 2, 2016.
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...and those are just the ones from this month. There are plenty more if you care to look through the past several months. So my best guess is that's where the game came from. I can't imagine that Amazon themselves would go out and buy UK versions and sell them to US customers. It has to be a 3rd party who is shipping them into the warehouses and representing them as US versions.