This is a 32X romhack, correct. If you run it from a flash cart (like the Everdrive) and also have a Sega CD attached, then that enables the other features.
32X Doom tried to be like the original PC game, but it was rushed through development as a launch title with a small team. The result is…tragic. Specifically: half the maps were missing, both the game window and resolution are reduced to get playable framerates, the original soundtrack famously sounds a bit like a farts at times, several visual effects like parallax texture scrolling, transparency, and lighting effects are gone. (there’s a room in E1M5 near the end that the lights alternate between on/off, but the monsters on 32x are always visible.) This also means no invisibility power-up, or Spectre monsters.
My favorite jank is after the credits when you finish the game, it dumps you into a fake DOS prompt. It just shows C:\DOOM> and you are unable to interact with it in any way.
Retro-bit, Retro Fighters, and Krikzz are a few more manufacturers I can think of making controllers with original ports.
In particular, Retro-bit’s Saturn Pro pad is…interesting, let’s say. I have also used Retro Fighters Striker Dreamcast pads - they’re quite nice.
In yonder days, a few companies like ASCII and Hori come up a lot for reliable stuff.