If we go by what Doomsday Clock has established...no, not exactly.
Superman, for instance, would probably remember lifting up that green car in 1938, being active during the WW2 years, and all the other events of the actual Golden Age comics up till the mid 1950's. But he probably wouldn't remember stuff like marrying Lois Lane, Power Girl being his cousin, surviving COIE in the paradise dimension, dying after IC etc. because all that happened to the 'Earth Two Superman'.
The way Doomsday Clock sets it up is that there is a mainstream DC earth who's history keeps getting rewritten, and every time history is rewritten, a new parallel earth is born 'preserving' the old history. When Superman's history was altered in the transition from the Golden Age to the Silver Age, a new version of the Golden Age Superman was created on Earth 2...and that's the Kal-L we know who was involved in COIE and ultimately killed by Superboy Prime at the end of IC.
Similarly, in the case of Kara, she'd probably remember being the Silver Age Supergirl and dying during COIE. But she wouldn't remember being Power Girl because that's a parallel universe version of her, not her in a 'past life'.
That may well be the case. My point is that the Omniverse allows for there to be multiple variations of even the same universe/continuity, to give the different divisions creative freedom if they so desire.
For instance, say a DC film wants to revisit the Reeve/Donner version of Superman. They are free to do so without being beholden to what the Arrowverse COIE did with that version of the character, as played by Brandon Routh.