The same “me” persists over time. The “me” that I vividly remember being at my first middle school dance is the same “me” that is typing these words right now. I’m older and wiser than I was, but it’s one and the same person. I can “re-live” those memories because the self is a real entity that persists over time.
Quite so. Intuitively and phenomenologically, that is your sense of self or personal identity and that it “persists over time”.
However, the philosophical problem of personal identity is complex.
From the wiki on personal identity
In philosophy, personal identity refers to the unique identity of persons through time. That is to say, the conditions under which a person is said to be identical to himself or herself through time.
Firstly, the mind-body problem:
The mind-body problem concerns the explanation of the relationship, if any, that exists between minds, or mental processes, and bodily states or processes.
Next, consciousness and personal identity:
According to Locke, personal identity (the self) “depends on consciousness, not on substance” nor on the soul. We are the same person to the extent that we are conscious of our past and future thoughts and actions in the same way as we are conscious of our present thoughts and actions.
The bundle theory of the self:
Hume, similar to the Buddha, compares the soul to a commonwealth, which retains its identity not by virtue of some enduring core substance, but by being composed of many different, related, and yet constantly changing elements.
The no-self theory:
Another view of personal identity is known as the no-self theory. According to this view the self cannot be reduced to a bundle because the concept of a self is incompatible with the idea of a bundle. This is because the idea of a bundle implies the notion of bodily or psychological relations that do not in fact exist.
Hume and the Buddha wrt the no-self theory:
On Giles’ reading, Hume is actually a no-self theorist and it is a mistake to attribute to him a reductionist view like the bundle theory. This reading is supported by Hume’s famous assertion that personal identity is a fiction. On this account the Buddhist view of personal identity is also a no-self theory rather than a reductionist theory. This is because the Buddha clearly rejects all attempts to reconstruct the self in terms of consciousness, feelings, or the body.
Personal continuity is an important part of identity; this is the process of ensuring that the quality of the mind are consistent from moment to the next, generally regarded to comprise qualities such as self-awareness, sentience, sapience, and the ability to perceive the relationship between oneself and one’s environment.
Buddha’s paradoxical middle way:
The Buddha attacked all attempts to conceive of a fixed self, while stating that holding the view “I have no self” is also mistaken. This is an example of the middle way charted by the Buddha.
Free will model of consciousness:
In the Freewill model of consciousness the brain models its own unconscious processes just as it models other people. This modeling makes the assumption that the model will continue to apply through time, and so assumes they are the same person they were yesterday. This leads to the intuitive sense of self.
The problem is, what is consciousness remains, because it cannot be assumed that the brain, by modeling its own unconscious processes, “somehow creates” consciousness, the sense of self and persistence through time in the Freewill model of consciousness, which is a physicalist reductionist theory.