Doom ran on MS-DOS, which - by modern standards - provided a shockingly minimal set of abstractions for programs. I think about the only thing you need to run Doom is the int 21h "API" to access the FAT filesystem and perform keyboard I/O. Note that MS-DOS did not provide facilities such as memory virtualization / management, process management, video drivers, sound drivers, etc - that was all provided by the hardware itself, which had its own hardware interrupts handled by the code in the device's ROM. It's why Doom required you to choose the type of a sound card you have, the interrupt / DMA channel to use, etc.
So I think this is a lot less of a flex than it seems; in fact, Anthropic using agents to build a semi-unusable compiler is far more of an achievement. Providing enough of int 21h to run Doom is probably something that a human could do in a weekend, doubly so if they can peek at the source code of FreeDOS.