See “rationale” in the original proposal
https://wiki.openstreetmap.org/wiki/Proposal:Tag:highway%3Dbusway
My understanding is that busway was intended to fill a perceived gap in the overall transport infrastructure tagging, for infrastructure related to BRT systems. I suspect it makes intuitive sense if you live in a city where BRT is a key part of the infrastructure, including several large South American cities. Short bus roads connecting to depots and so on were already covered by highway=service.
I think this brings us back to the “duck tagging or not” discussions referred to on several other threads recently. Is highway=cycleway for any random bit of path that a government decides to stick a blue bicycle sign on, or for actual cycleway infrastructure? Is highway=busway for any random piece of road reserved for buses, or for actual BRT corridors? (Perhaps brt_way would have led to clearer duck tagging but the proposers might have been told off for inventing English words).