performance differences come down to how the driver utilizes the firmware, or how the applications utilize the driver. there may be some other factors, like how the OS manages memory buffers.

it's unlikely that the firmware has OS-specific code. it's more likely that the firmware exposes functionality that happens to be taken advantage of by one OS' drivers more than it is by another OS' drivers -- perhaps in part because of differences in driver execution models on different kernels. or sometimes (as with nvidia) because a proprietary closed-source windows driver was written by the company with access to private documentation of all the firmware's features while the community-written linux OSS driver was written with incomplete knowledge of the firmware's features derived from reverse engineering.

Performance-wise yes, logicaly the difference would come from the driver part.

For the firmware, it's hard to know, i think only someone from AMD can answer this. It's technically possible if part of a firmware is used only by the Windows driver for example. After, I have no idea, if it's the case...