Even though Jeremiah acts as Gotham's precursor to the Joker, he will never be called "The Joker" as Cameron Monaghan confirmed on his Twitter account due to Warner Bros. wanting to use the name "Joker" for future projects. Monaghan also confirmed Jeremiah cannot have bright green hair.
When Jeremiah is undergoing his villainous transformation, he clutches his head in a way that mirrors the famous panel of The Joker going insane in Batman: The Killing Joke.
When Gordon shows up to Jerome request, his Arkham Asylum Lunatics are playing the theme to Batman (1966), including the lyrics "Batman! Batman!"
Because it's Bruce Wayne's birthday, the date is February 19th.