Although Barry Keoghan is simply credited as "Unseen Arkham Prisoner," it's obvious that he is the Joker. Director Matt Reeves has revealed that in The Batman's universe, Keoghan's Joker has had his scars from birth and his permanent smile is the result of a congenital disease, so he can't stop smiling. Batman has already encountered Joker in his first two years and they have a history together, although he isn't calling himself Joker yet. The obscured makeup was based on Conrad Veidt's appearance in The Man Who Laughs (1928). Veidt's role in this movie was the main influence for writers/creators Bill Finger, Bob Kane and Jerry Robinson to create Joker.
Three weeks after the film's release, Warner Bros. released a deleted scene online where Batman visits an inmate in Arkham State Hospital. During the scene, Batman interrogates him about who the identity of The Riddler possibly is. The scene was well received, and many people wondered why the scene wasn't in the film. Director Matt Reeves stated that he deliberately left it out because the scene tells the viewer exactly what motivation the Riddler is acting on.
When Batman consults with the Joker in prison over the Riddler murders, he hands him a file with a paper clip on it. When Joker hands back the file, the paper clip is gone, likely taken by him for use as an improvised lockpick to escape.
Included as a special feature (with optional commentary by director Matt Reeves) on bluray releases of The Batman (2022).
Batman consulting the Joker on the case is a reference to Manhunter (1986), where Lecktor similarly gets inside Will Graham's head with only a few choice words. It even has Bruce offering to let him read the case file (with photographs, which Joker shows interest in) as incentive to help, and handing him the documents through a sliding tray in the partition.