Warning: The following contains SPOILERS for Legends of Tomorrow season 6, episode 10, "Bad Blood."
Legends of Tomorrow season 6 established that the Justice Society of America (or JSA for short) exists in some form on Earth-Prime. This is notable, as there had been no previous indication that the team still existed outside of the new Earth-2 of Stargirl following Crisis on Infinite Earths. This begs the question of how it is possible for the JSA to exist in a publicly recognized form when there is nothing in history to suggest openly operating costumed vigilantes during World War II in the Post-Crisis reality of Earth-Prime.
The main plot of the Legends of Tomorrow episode "Bad Blood" found warlock John Constantine and alien-hunter Spooner Cruz taking a trip back in time to 1939 and the town of Albacete, Spain, in search of a mystic artifact called the Fountain of Imperium. While there, the two befriended a mute teenager named Fernando, who loved comics and thought the pistol-packing Spooner was La Torera; a masked heroine similar to Zorro. Later in the episode, after they were taken hostage by Spanish Nationalists who were also searching for the Fountain of Imperium, Spooner tried to comfort Fernando by telling him that real superheroes would be coming to their rescue. This led Fernando to inquire, "Like La Torerea and (the) JSA?"
The Legends of Tomorrow encountered a version of the JSA in the past, but that occurred before Crisis on Infinite Earths, with little indication it still existed in the revised Earth-Prime timeline. Even if they did, this JSA had been a secret group organized by the US government, so there was no plausible way a teenage comic reader from Spain could have known of their existence. It's even more unlikely that he would somehow know of the JSA unique to the reality of Stargirl. Ironically, there is an explanation that clarifies the confusion of the different JSA teams, but it lies in the comic books of our Earth and the story that first introduced the DC Comics multiverse.
The Pre-Crisis Arrowverse Earth-1 JSA
The first mention of the Justice Society of America in the Arrowverse came in Legends of Tomorrow's season 1 finale when the Legends were confronted by a hooded man who crashed a duplicate Waverider next to their ship. The man introduced himself as Rex "Hourman" Tyler and a member of the Justice Society of America, claiming to have been sent back in time by a future version of Mick Rory to stop the Legends and their leader, Rip Hunter, from getting on board their ship and traveling to their deaths. Legends of Tomorrow season 2 introduced the rest of Hourman's team, including Stargirl, Doctor Mid-Nite, Obsidian, Commander Steel, and Amaya Jiwe.
This version of the JSA was a covert organization, operating in secret to take on those missions that ordinary American soldiers couldn't handle during World War II. Most of the team died in the line of duty, after being recruited by Rip Hunter to take the fragments of the reality-altering Spear of Destiny and hide them across the timeline. This was why, in the original timeline of Earth-1 before Crisis on Infinite Earths, there was no record of costumed heroes existing before Oliver Queen became Green Arrow and Barry Allen became The Flash. It should also be noted that there is no indication that this team existed in the Post-Crisis timeline of Earth-Prime.
The Post-Crisis Arrowverse Earth-2 JSA
A new multiverse was formed following Crisis on Infinite Earths, with the worlds of Earth-1, Earth-38 and the Earth of Black Lightning merging together into a new world called Earth-Prime. A new Earth-2 was also formed, which became the setting of the Stargirl series. The Justice Society of America was the first superhero team in the history of Earth-2, coming together during World War II and remaining active until most of the team was killed in a battle with their archenemies in the Injustice Society on Christmas Eve 2010. Courtney Whitmore, who adopted the name of Stargirl after reawakening the Cosmic Staff of Starman, formed a new version of the JSA after the fight against the Injustice Society. This new team operated in secret, like the Earth-1 JSA, but the original Justice Society of America and their individual members were frequently profiled in the newspapers of the time, according to Pat Dugan. Until Fernando's mention of the JSA in Legends of Tomorrow season 6, there was no indication of a similar team operating on Earth-Prime following Crisis on Infinite Earths.
"The Flash of Two Worlds"
A classic issue of The Flash comic may offer an explanation for how the Justice Society of America can now exist on Earth-Prime, despite apparently being shuffled off to the new Earth-2. "The Flash of Two Worlds," which was originally published in The Flash #123 in September 1961, detailed the first encounter between the Silver Age Flash, Barry Allen, and the Golden Age Flash, Jay Garrick. Having reportedly been asked what had happened to the original Flash in the years following the introduction of Barry Allen, writer Gardner Fox decided to answer the question with a story that introduced the concept of alternate realities into DC Comics' cosmology, with Barry Allen living on Earth-1 and Jay Garrick living on Earth-2.
The story detailed how Barry Allen accidentally broke through the barrier between worlds and wound up on Earth-2 in Jay Garrick's hometown of Keystone City. This astonished Barry, who knew everything there was to know about Jay Garrick and his secret identity, having been a fan of The Flash comics as a kid and taking inspiration from Garrick when choosing his own superhero codename. When Jay Garrick asked how Barry had read about his adventures as a child, Barry theorized that the writer must have somehow "tuned in" to Garrick's Earth and transcribed the events of Jay's life thinking he was imagining the stories himself.
Theory: The JSA Exist In Earth-Prime's Comic Books
Taking "The Flash of Two Worlds" into account, it seems likely that Justice Society of America exists on Earth-Prime as fictional comic book characters. While it is possible that the team also existed in some covert capacity (like the JSA of Earth-1 seen in Legends of Tomorrow season 2) it seems improbable that the American government of that time would allow a publisher to print books based on a team that was meant to be secret. It also seems likely, given that most of the writers of the Arrowverse love sneaking nods to the classic comics into their scripts, that the mention of the JSA was an intentional reference to "The Flash of Two Worlds," which had been previously referenced several times on The Flash. The simplest and most logical explanation for Fernando's being a fan of the JSA is that some Earth-Prime comic book writer has developed a psychic connection to Earth-2 and seen the adventures of the Justice Society of America play out in their dreams.