Technically they would have drivers but it's such a standardised interface that it is all simply built into the operating system. It wouldn't even provide a mechanism for replacing the driver hence it would not appear in, say, Windows' device manager.
USB is a slightly different situation because a USB connection can carry many different types of device interface, including hubs which in turn can carry other devices and other hubs. Also, USB has a little more overhead because your system has to be able to determine things like what type of device it is, how much power it needs, etc before it can run, so you don't just get instant usability, you have to wait until the OS has time to probe and initialise the device. Nonetheless, most such device interfaces still have support included in the operating system and have no requirement to install a specialised driver - for instance, keyboards, mice and external hard drives. Windows, annoyingly, will go through the process of "setting up a new device" when you insert a new device even if it has the required drivers already because it's just a standard device.
Both USB and PS/2 devices can have manufacturer-provided drivers, and these are often optional in the sense that all they do is provide some sort of additional features related to non-standard aspects of the device.
The advice about trying a PS/2 keyboard if your USB keyboard is not recognised by the OS is simply a way of troubleshooting if there is something wrong with your system's USB support. There are various things that can go wrong with USB. Firstly, USB support is more complicated so some boot managers or even "BIOS" setup interfaces don't support it, or do but only when your system is set to "legacy" USB support. It's also possible to disable USB support sometimes, and re-enabling it may be difficult without a PS/2 keyboard. And sometimes there are other issues with USB.
Additionally, it is my understanding that PS/2 support is legacy support that is baked into the BIOS on most systems. Not a PC guy, but I assume EFI/UEFI systems similarly have basic PS/2 support in place but the adoption of EFI/UEFI is a relatively recent phenomena that only goes back as far as 2011. – JakeGould – 2018-11-27T00:10:18.607
@JakeGould As far as Windows is concerned it doesn't matter if support is "baked into" the BIOS or UEFI. Windows uses the BIOS or UEFI only until WinLoad has been loaded (very early in the boot) - after that it uses its own drivers. – Jamie Hanrahan – 2018-11-27T08:31:12.937