I think a considerable subset of the people who do make use of the Chinese room argument also subscribe to some form of mind-body dualism, where consciousness does not or does not completely arise from physical processes.
The Chinese Room and the brain of a Chinese-speaking person are completely different physical processes. Looked at on an atomic level, they have almost nothing in common. Mind-body dualists may or may not agree that the room is not "conscious" in the way a human is, but if consciousness is purely a material process, I can't see how the materialist can possibly conclude all the relevant properties of the completely dissimilar room and person are the same.
Those that would argue the Chinese Room is "conscious" in the same way as the Chinese person are essentially arguing that the dissimilarity of the physical processes is irrelevant: the "consciousness" of the Chinese person doesn't arise from molecules bouncing around their brain in very specific ways, but exists at some higher level of abstraction shared with the constituent molecules of pieces of paper with instructions written in English and outputs written in Chinese.
The idea our consciousness exists in some abstract sense which transcends the physics of the brain is not a new one of course. Historically we called such abstractions souls...
The obvious counterpoint is that if I followed your argument to absurdity, then I would also have to conclude that if I am conscious, then you can't be, because the atoms of our brains aren't arranged in precisely the same way. It clearly makes more sense from a monist point of view to consider consciousness an emergent property of complex systems, rather than one particular process.
