The idea that the "self" is a process rather than a thing, which is basically what this claimed "new scientific discovery" amounts to, is actually not new to cognitive science. But it's also not the same as claiming that the "self" doesn't exist. Yes, there is no particular set of neurons in the brain that corresponds to the "self". But the whole process that your brain and body are constantly executing is a reasonable scientific embodiment of your "self".
To the extent there is an Eastern philosophy counterpart to this, it is the idea that it is wrong to pick out one particular part of the whole process going on in your brain and body--say the part that constructs a "narrative" that you can speak, either to yourself or to others--and treat that as though it were something separate from the rest. Much of the teaching of, for example, meditation can be seen as simply trying to make you more aware of all the other processes that are just as much a part of your "self" as the "narrative" process. But this article is taking the opposite view: that the only candidate for your "self" is the narrative process. I don't think that is a valid reading of Eastern philosophy any more than it is a valid conclusion from the science.