Does a reincarnated person have memory of their past lives?

In Buddhism the world you conceive & perceive is a world in a system of worlds.

Whatever in the world through which you perceive the world and conceive the world is called the world in the training of the Noble One. And through what in the world do you perceive the world and conceive the world?

Through the eye in the world you perceive the world and conceive the world. Through the ear … nose … tongue … body … mind in the world you perceive the world and conceive the world. sn35.116

The entirety of a world is conceived & perceived by this or that nervous system and there is no conception nor perception of the world otherwise.

Each frame of reference wherein conception & perception occur is a closed system of information, no person can experience the world as another and thus no two conceived & perceived worlds are the same because their conception depends on two different instances of sensory contact.

A person's mentality such as feeling, attention & perception, are dependent on this or that physical frame of a body but this body is just a small part of a world conceived by this or that person.

As wherever you are is a "here" for you and you can be anywhere in this world you perceive whilst still being "here", the "here" changes as it persists and it is not one and the same "here" but they are all of this world. As time is always "now" and as this "now" changes as it persists, it's not the same "now" but it's in the same world which also changes as it persists.

So does there being a body, in dependence on which there is a conception & perception of this or that world in a world-system, persist while it changes in course of transmigation. Next life is simply the next world, next here, next now, next body, next mind but still in this in the very world system.

Maybe you can see that it follows that any one capable person having conceived a world, has with that conceived the entirety of formation & information of that world and can access any information without any storage of memory other than that which is that world system conceived by a mind which also which changes as it persists through transmigation until the extinguishment of that which changes as it persists for a cessation of the world.

The Blessed One said: "And what is the origination of the world? Dependent on the eye & forms there arises eye-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. This is the origination of the world.

"Dependent on the ear & sounds there arises ear-consciousness. The meeting of the three is contact... Dependent on the nose & aromas there arises nose-consciousness. The meeting of the three is contact... Dependent on the tongue & flavors there arises tongue-consciousness. The meeting of the three is contact... Dependent on the body & tactile sensations there arises body-consciousness. The meeting of the three is contact... Dependent on the intellect & mental qualities there arises intellect-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. This is the origination of the world.

"And what is the ending of the world? Dependent on the eye & forms there arises eye-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. Now, from the remainderless cessation & fading away of that very craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging & death, sorrow, lamentation, pain, distress, & despair all cease. Such is the cessation of this entire mass of stress & suffering. This is the ending of the world.

"Dependent on the ear & sounds there arises ear-consciousness. The meeting of the three is contact... Dependent on the nose & aromas there arises nose-consciousness. The meeting of the three is contact... Dependent on the tongue & flavors there arises tongue-consciousness. The meeting of the three is contact... Dependent on the body & tactile sensations there arises body-consciousness. The meeting of the three is contact... Dependent on the intellect & mental qualities there arises intellect-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. Now, from the remainderless cessation & fading away of that very craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging & death, sorrow, lamentation, pain, distress, & despair all cease. Such is the cessation of this entire mass of stress & suffering. This is the ending of the world." sn12.44

In other words whatever is grasped with wrong view as or as personal for this or that person, mental or physical, is part of nature conceived & perceived by that person and being part of nature all information about that nature is accessible right there. It is as if asking if nature's intellect can know that which can be known of nature. There is simply nothing else that can know it other than intellect which develops in that very nature and it can be grasped with wrong view to be personal or as a person.

Not all people can recall their past lives. Only some small percentage can or unless one's an advanced meditator. However, it's important to notice that Buddhism doesn't really have the concept of a "soul", meaning there's a perpetual unchanging "self" entity transmigrating from life to life. So although there're rebirths where one's Kamma serves as the fuel/propellant, there's no permanent unchanging "self" associated with it. The expression "I" in the sutta excerpt below simple serves as a label to describe the new manifestation of Kamma in a new person with a new identity in a next life, not a permanent unchanging "soul":

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of recollecting my past lives. I recollected my manifold past lives, i.e., one birth, two...five, ten...fifty, a hundred, a thousand, a hundred thousand, many eons of cosmic contraction, many eons of cosmic expansion, many eons of cosmic contraction & expansion: 'There I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose there. There too I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose here.' Thus I remembered my manifold past lives in their modes & details. ~~ MN 36 ~~