edit: And how could I forget (as mentioned by another poster in the thread) the numerous NEC PC98 machines with an 8086 which created their own industry. Sorry for slight, Japan.
https://www.minuszerodegrees.net/misc/Determining%20Processo...
With the exception of bus speeds / timings and the differently sized prefetch queue, the 8088 and 8086 are 100% software-compatible so there wouldn't need to be any difference in the BIOS.
By the way, while the PS/2 series marked IBM's first use of the 8086 in a PC (specifically the PS/2 Model 25), clones had been using the 8086 for some time already. The 8086 worked seamlessly with the standard ISA bus, and as far as I know, no 8086-based systems used the PS/2-specific Microchannel bus.