Saber remembers stuff in FSN and all because she's still alive and the last summoning "just" happened, but she also experiences them more like dreams than actual memories, so its ambiguous as to how much exactly she'd retain after death.
FGO just throws in fanservice lines in my room because its amusing. Her interlude in Fuyuki implies some familiarity, but not actual like, "Oh yes, this is the place where I met the Emiyas and hated one and maybe fell in love with the other."
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