This hypothetical person regards the 'self' as the individuality physically distinguishable -- the physical body composed of the elements, born from mother/father.
Then, arguably such 'self' as used by this person might be the same 'self' as used in conventional language by the Buddha and the Arahats. The word, then, could have the same function for both groups, in this context.
While such person likely denies the existence of an enduring self -- which would be in agreement with the Buddha -- the quote suggests the view that death marks the end of experiencing, that death is the end of experience. And this, according to the Buddha, is wrong view.
The best argument I know of in the suttas for this stance is that, if such was the case, then karma (and, consequently, morality) would be fiction: there would be no results of actions according to their moral nature (and consequently, morality would be an unnecessary obstacle to gains and benefits, to happiness). Or, more precisely, such results could only occur during this very life -- so one better do all evil things as fast and careful as possible and enjoy the benefits of it while avoiding any bad external outcomes (outcomes that are not really governed by karma law, but exclusively by social conventions of a given time and given culture, e.g. criminal laws and law enforcement).
Another argument against this view can be inferred from the suttas around Nirvana. That is, under this view, Nirvana would have a doubtful value. Why would anyone spend a lifetime avoiding worldly pleasures, enduring hardship, for the promise of an hypothetical "best pleasure" that would only last for a few years? That is, until the person dies. And yet, such person would still suffer physical pain, just like anyone else. Under this view then, once he/she dies, it doesn't matter who attained Nirvana and who didn't, since the same thing happens to everyone equally: the end happens.