I wouldn’t say so at all. We have a VERY VERY limited understanding of consciousness. But I think that is it fair to say that, within this understanding, copying a brain perfectly does not result in one consciousness sharing two brains but two consciousnesses in separate brains. This need not even involve an understanding of consciousness, the two brains would somehow have to be connected to share consciousness unless you believe in the supernatural/spiritual (we’re assuming you don’t because that’s hogwash), and there’s no logical way for that to occur simply from the processes of copying. Therefor the rest of the thought experiment should stand.
It gets very interesting when, if you consider the thought experiment valid, you consider another one in which only parts of the brain are copied, and then they are used to replace the part they were copied from. You do this one at a time until you have an entirely copied brain in the original skull. When is consciousness lost/switched? Is it? If it isn’t lost/switched, then what size part can you replace before it is? Would something more complicated like a gradual metamorphosis occur to the consciousness? I don’t think I can provide any answers.