I've read that the Buddha could remember his past life. Does this means that the memory of a person lasts across different life times?
This is just knowledge (jnāna or vidhyā). Anyone who acquired that knowledge can remember his past life.
Please see Buddha's awakening Buddha's awakening
and page 63 of "On Being Buddha: The Classical Doctrine of Buddhahood"
and I retrieved the text below from page 571 of "Agama Aura Tripitaka: Eka Anusilana"
Prodigy of Memory of Earlier Births (Pϋrva-nivāsa-anusmrti-jnāna): The monk endowed with this prodigy remembers many kinds of earlier births. He can remember many periods of duration of dissolution, creation and destruction-cum-origination of the universe. He remembers that at that time, he was in a particular place, state of birth, species, destinity and state of knowledge. I was in a specific state of birth. I had this name, lineage, life-span, food and experience of pleasure and pain. I was born in a specific place after death from there where I had this…etc. After death there, I am born here in this Ksatriya or Brahmin family. The heretics remember their earlier births upto 40 Kalpas (a unit of time equivalent to 20 Antar-Kalpas). The natural votary remembers earlier births upto 100 or 1000 Kalpas, the great/senior votaries remember earlier births upto 80 lacs of Kalpa years, the self-enlightened one remembers the earlier births upto two innumerable lacs of Kalpa years and the Buddha remembers the earlier births upto a period without measure.
If I explain in simple way ,When we see it out of the box even though it is like our memory is saved in somewhere it is not like that.. Once we improved our mind using bawana ,widharashana ,...etc we would be able see this world in ever most realistic way..That is really strengthen of the ultimately well improved mind.It is called "nyana" ,once Buddha acquire his goals he was gain mainly three special "nyana". Using them he was be able to see past of animals(all sort of lives) ,as well as future of animals and there is other one. Really that's why we address he is "Buddha"there is nothing to realize beyond his understanding in this world ever. As we are not having such a biggest improve in our mind, worldling scopes are very narrow .Ex: think about human eye, human would be able to see withing specific spectrum(between range of frequency)beyond this spectrum there are a lot of things.But we can't see.But sometimes other animal can see different things due to that they belong to different spectrum. As same as our mind is like that.But the special thing is, if we can improve our mind we would be able to expand this range unless eye,ear or other part of our body. That's what Our Buddha ultimately retrieved.
Lets take our current life to stat with. We do not remember everything. We have few memories of what we did as a toddler. Next to none of our 1st year and when we were in the womb. Our memory fades. Very few people just remember their past life.
By shaping our mind we can start to see back into the future remembering more of things, especially if you develop Jhanas.
Since at the point of death and rebirth there is a strong break from the material body remembering things beyond is difficult without Jhana through there are some who can do it.