Starting in 1966, DC told us that there were two Supermen and two Batmen. A radical idea. The path had been prepared earlier with versions of (importantly) Flashes and later Wonder Women of two worlds. And so a seed of confusion was planted.
Just to throw out an approximate breakdown of the two DC flagship heroes' histories by era. Plenty of room to pick nits and further subdivide considerably... this is just an approximation.
A 1939: The original, darker Batman
B 1940-1952: The somewhat lighter Batman paired with Robin; referred to in JSA stories
C 1953-1963: The much lighter Batman with Batwoman, Bat-Mite, sci-fi galore; shares universe with Superman and Club of Heroes but no JSA or JLA (until 1960)
D 1964-1971: The New Look Batman
E 1972-1986: The more gothic Batman, including Englehart & Rogers' run
F 1987-2005: post-DKR, "Batgod", "Bat-jerk"; many retcons, some undoing other retcons
G 2006-present: Morrison [Dini] Batman
H 1966-1979: Earth Two Batman with JSA past, married to Selina, killed by Bill Jensen (referred to since his death many times in the past tense)
A 1938-1942: Superman based in Cleveland, works for George Taylor at Daily Star; jumps but cannot fly
B 1943-1952: Superman based in Metropolis, works for Perry White at Daily Planet; flies and is much more powerful; referred to in JSA stories (1940-1947)
C 1953-1960: Superman folkloric in power, shared universe with Batman and Robin; past as Superboy (1945-1986)
D 1960-1972: first and most powerful in a world full of superheroes
E 1973-1985: post-Sandman Saga; numerous plots questioning and decreasing the primacy of Superman and JLA while retaining history of Silver Age
F 1986-2003: Byrne and Triangle Era Superman retelling many elements of the previous history
G 2004-2006: Birthright / Byrne-Birthright Superman
H 2006-present: soft reboot to more Silver Age blend of post-Crisis Superman
I 1969-1986: Earth Two Superman, member of JSA, contradictory accounts of power levels, married Lois
J 2005-2006: Kal-L escaped from paradise dimension, had crossover with post-Crisis Superman, killed in Infinite Crisis; said to be same as (I)
Now... in almost all cases, the Superman-Batman histories in the main comics are an unending affirmation of the previous history. The transitions are almost always presumed to maintain the previous history except where otherwise noted. The ONE exception to that is the introduction of the Byrne Superman, which came with the understanding that the previous history was not to be presumed until we'd seen in directly. Batman's (C) era was almost totally ignored by (D, E, F); not explicitly denounced, but definitely excluded from those eras by tone.
The Earth Two Batman (H) and Superman (I+J) were supposed to have begun their careers with the respective (A) versions, just as seen in the original publications, and set pre-WW2.
Batman, unlike Superman, has twice had his oldest history reaffirmed. (E) and (G) took stories from era (A) and placed them back into his history. Definitely with the presumption that they took place about 10-15 year earlier than the cover dates, not in 1939.
The "Golden Age" (a term coined by a fan, remember) corresponds to eras of publication (A) and (B) with no definite ending.
Now, when the Earth Two characters were brought into JLA-JSA crossovers, there was a bit of a white lie being told. The monthly comics had never introduced any sharp break in history. (The 1942ish transformations of Superman and 1964 introduction of the New Look Batman came closest.) And the Batman and Superman solo titles had made no mention of the other superheroes until they "met" in 1952, even though JSA stories referred to (and rarely included) Batman and Superman. Given that Batman #13 asserted that Superman was a fictional character even though they had been included in the JSA's "universe" as of 1940, you could argue that the Batman titles, the Superman titles, and the JSA amounted to three different continuities.
To say that the Earth Two characters were the Golden Age characters was a bit of a white lie. All of the sharp identity changes supposed to exist between the Earth Two and Earth One Supermen happened before Superman's one active JSA adventure in 1947. The JSA Superman, at least in his solo titles at that time, was the son of Jor-El, could fly, had been Superboy, and worked for the Daily Planet in Metropolis under Perry White. How do you like them apples? While your head is still reeling, grapple with the fact that the first appearance of the Earth Two Batman showed him alongside the JSA with a yellow oval on his chest with Blockbuster, a character who debuted five years after the JLA was begun. Two worlds couldn't be more mixed up if you saw the Pope celebrating Hanukkah.
So the "Earth Two Superman" (I) was a brand-new fusion of something that had never been: a guy who started off like Superman (A), but stayed that way through a career during the heyday of the Justice Society. I think owing to the extremely greater fan awareness of Superman's (A) era than his (B) era, this white lie was easily swallowed by fans from 1966 to the present. But in no way except one was the Earth Two Superman the Golden Age Superman any more than Englehart's Batman was the Golden Age Batman. And that one way is: The calendar of his publication and his supposed history were the same: He was supposed to have operated before and during WW2. Whereas Englehart's Batman had the history of Batman #1 in his past, you would have to assume, in 1978, that those events would have happened around 1970, earlier in his adulthood.
So this use of "Golden Age" to refer to the Earth Two characters is a bit of a lie for Superman and Batman. We can say that we know what DC means when they use the terms that way, but when it comes right down to it, the introduction of a second Batman and a second Superman was just the further tangling of a web that was already pretty well twisted by the minor inconsistencies that had grown over the preceding 25 years.