If I recall Saber pretty much says fuck you to that with some of her lines
Pretty much all the Servants who can be summoned who were summoned in previous on-screen HGW do the same thing. Saber's a special exception because she was technically alive during the two we saw her in, and not a proper Heroic Spirit, but pretty much everyone has recollections. Atalanta explains in her second interlude that in most cases, summonings become almost like a dream that fades away. Living so many "lives" becomes almost unreal. But powerful enough experiences stay with them... which conveniently could be said to describe pretty much every on-screen HGW we've seen thus far. For example, memories of her encounter with Jack the Ripper in Apocrypha haunt the FGO summoning of Atalanta, in both her base and Alter forms.
That being said, there are still Servants who treat these past summonings as independent entities from their current summoning, notably Semiramis and Amakusa Shirou.
The Hassans are another exception, but that has more to do with the concepts of wraiths as opposed to Heroic Spirits.