PadmaVonSambha wrote: a person doesn't die and that same person takes rebirth somewhere else.
although this is the usual way it is expressed, for the sake of convenience.
That is a Hindu concept based on the notion of an eternal soul.
Buddhist rebirth is a different concept,
A whole different process, involving uncountable mental components that are experienced as a single "me".
But I don't have time to go into more detail than that.
Allow me, then. The differences between Hindu~Vedanta and Buddhist concepts are not nearly so great as the differences between them and (for instance) the Semitic religions. They both developed against the background and acceptance of the 'eternal round of birth-and-death' from which release can only be won by vidya or spiritual insight. Of course there are differences as to what constitutes vidya and the nature of the path (margha) but the similarities are still profound. The relationship is further complicated by the fact that through millenia of debating, Vedanta and Buddhism have profoundly influenced each other. (Sankara was accused by more orthodox Hindus of being 'Buddhist in disguise'. I tend to think Buddhism influenced Vedanta more than vice versa, but it's a contested point. Richard Gombrich's What the Buddha Thought and How Buddhism Began are good on the similarities and differences between the Hindu and Buddhist views and the degree to which the latter diverges from the former.)
It is also true that the Buddha doesn't accept the idea of a 'permanent unchanging entity'. But in his view, it is not only persons who are impermanent - but everything! There is no permanent unchanging thing anywhere to be found. However interpreting the meaning of this is quite difficult in my view. According to another Buddhist studies scholar (Stephen Collins in Selfless Persons) the 'polemics of anatta' are indicative of a certain style of teaching within which the notion of 'self' became 'a linguistic taboo in technical discourse' - mainly because of the constant necessity of differentiating the Buddhist approach from its competitors, particularly the Vedanta. But this lead, in his view, to a lot of sophistry and sometimes even double-talk, insofar as it requires the explanation of karma with no agent to whom it accrues.
If you look at Tibetan Buddhism with its ideas of 'voluntary rebirth' and the traditional means of identifying incarnate lamas by recognizing objects from their past lives, and so on, I think it is hard to maintain the artifice of there being no personal continuity. What has happened, in practice, is that it is simply re-conceptualized as 'a mind-stream' instead of 'an entity' - but in practice, it adds up to the same thing.
My view is that the realization of 'no-self' (which is really another facet of the realization of emptiness) is not a proposition of whether or not people 'have' or 'are' souls or whether anything is eternal or not. It is, I think, more a matter of the being realizing its non-affinity with anything in the 'realm of sense' whatever and so waking up to its 'true nature', however that is conceived.