Senshin is correct, but there's another example to point out.
Senshin is correct that heroes from the throne do not remember individual summons (for any purpose), as the copy that is stored in the throne is fixed. There is, however, a mechanic which can generate exceptions: reality marbles. The stored hero and his various summons all share the same reality marble. A reality marble, being somewhat literally the user's internal world (which is then made manifest upon the world in some sense and extent), changes in response to changes to that internal world. As such the reality marble becomes the channel through which memories and personality changes are conveyed and imposed upon the stored hero and subsequent summons.
This is how Archer becomes distorted and warped, and develops a plan to try to end his own existence by paradoxically killing himself should he ever be conveniently summoned into the grail war in some alternate timeline. He was frequently summoned as a pawn devoid of the ability to enact his own will to mop up the messes of humanity, and to kill and destroy as demanded. He had made his pact with the world thinking he would prevent suffering and tragedy and death, not simply be a murderous janitor that comes in after the fact. His reality marble effectively records these things and it results in twisting the personality of the heroic spirit.
So in such a case, yes, a spirit can remember "prior" summons. Though exactly how the "order" of summons gets determined when the throne itself exists outside of time, I don't know; clearly the Fuyuki HGW we see occurs after many other summons for other purposes.
In principle this could apply to any other hero with a Reality Marble, such as the 4th War's Rider. Indeed, some people felt they saw an adult Waver in his reality marble's army in the Fate/Zero animation, but this seems fan speculation at best and may have been someone else with a similar look.