The story of DOOM showing up on the SNES is a pretty wild one. id Software had absolutely zero plans to bring the game out on Nintendo's console, but that all changed when a package from Sculptured Software showed up at id's offices. Inside the package was a reverse-engineered version of DOOM that ran on the SNES. Sculptured Software had worked on this prototype on their own, completely for free, and sent it off to id with hopes of them picking it up for publishing. Thankfully id was impressed, and helped bring the port from prototype to reality.
In an interview with Shacknews, DOOM dev John Romero opened up about the path for DOOM to the SNES.
"They reverse-engineered it and sent it to us and we said, 'Um. Okay. We'll get this published.' That was probably one of the rare times when a company decided to do a port and then just did it for free. We didn't even think to do a Super Nintendo version of Doom. That hardware was absolutely not made for 3D. Super Nintendo was good for horizontal scrolling, parallax effect, all that stuff, but Mode 7 was as crazy as it got. We were really surprised. We didn't think the Super Nintendo could do it, but incredibly, that was so many people's first exposure to Doom."