I have a newbie question.

I know at very low level communication is done buy register r/w via certain bus.

I assume register maps are different for different brand peripherals of the same type, e.g., PCIe wifi cards from different manufacturer.

I assume OS or kernel talk to the peripherals at an abstract level.

And what the level is?
And what paves between this level and the lower level?

An example is Linux on two computers with same CPU but very different WiFi cards.

that is called device driver. The kernel sends a unified message to the driver, and the driver will translate it and forward it to the device.

Devices on the PCI bus each have unique vendor/model codes that determine which driver gets called. You can read the PCI config registers with lspci.

To put it another way, the device driver presents Linux with a generic abstraction of the device. "It's a network card." Linux calls the driver's various subroutines, also handing it the device-table entry. It's the responsibility of the (vendor and device-specific) driver to implement each call – to actually "push the buttons and flip the switches in the hardware" to make a particular device do its thing.

Devices uniquely self-identify themselves, so that drivers can be sure they're talking to a compatible device and so that Linux knows what driver to associate with each device. (Which beats the hell out of fiddling with DIP-switches like we had to do in the bad old days!)

"two computers with same CPU but very different WiFi cards"

Think of it as a two cars that are identical except for the tires. Once you install the tires to the rims and to the car they act the same.

If you ignore the networking part of the question it comes down to the ability to access a device on a bus.

And of course a LOT of devices are identical internally no matter what the manufacturer stamps on the case, which is why you'll find drivers that are based around "chip-sets" rather than specific manufacturer/model.