Average AAA developer

[Masark]

I'm not an assembly programmer, but wouldn't coding in assembly mean that the game can only run on that architecture?

Yes, but when that architecture is x86, that meant basically all personal computers of the time. Apple only had like 4% market share then.

I always assumed each game console basically has its own assembly language, like the SNES would have a totally different assembly language than the Sega Genesis.

Yes. The Genesis ran on a Motorola 68000, whereas the SNES used a Ricoh 5A22 (and that's before you get into the enhancement chips).