It varies wildly depending on what the story needs. Some series treat past-life memories as a complete personality takeover—the new character basically wakes up one day with all the skills, emotional baggage, and worldviews of their previous self. 'Mushoku Tensei' does this pretty literally; Rudeus isn't just remembering, he's actively integrating his past self's failures and knowledge into his new life. That's a heavy psychological burden, and the show leans into it.
Then there are others where memories serve more as a convenient cheat code. 'The Rising of the Shield Hero' gives Naofumi modern-world business sense, which changes how he operates in a fantasy economy, but his past life doesn't haunt him emotionally in the same deep-cut way. It's a tool, not trauma.
My favorite approach is the fragmented memory trope, where recall is triggered by specific sensory cues—a smell, a song, a location. It feels more realistic than a full data dump at birth. It also creates suspense. 'Fushigi Yuugi' played with this ages ago; the protagonist's memories surface slowly, altering her loyalties and decisions piece by piece. That gradual reveal mirrors how we actually remember things, I think.
Ultimately, it's less about the 'how' of the memories and more about what the narrative uses them for: character depth, plot convenience, or a mix of both.