SPOILERS for all ahead:

Can the Joker universe (Joker and Joker: Folie a Deux) and The Batman universe (The Batman, its upcoming sequels, and The Penguin TV series) be considered the same continuity in headcanon, even if not in reality?

The way they're structured seems like they almost could be in the same universe, and many people questioned if they were at some point before it was confirmed they weren't. Joker kind of acts as an origin story for the Batman mythos and his "Rogues Gallery" generally, not for any specific version of Batman, but it seems to connect quite well with Matt Reeves' The Batman: Bruce Wayne is a child in Joker, Harvey Dent is quite young in Folie a Deux and just had his villain arc set up, and the new version of the Joker we see in Arkham at the end of the second movie was also not much older than Bruce at the time, so that they could serve as villains for him once he grew up. And we seemingly saw a version of the Joker in Arkham at the end of The Batman; maybe that's him, or yet another person who carried on the mantle of the Joker that was established in the 2 Joker films.

I want to make clear that I'm not asking if they are in the same universe, I know they're not (officially)... nor am I asking if they should cross over in some way, I don't think they should. Just whether they could be compatible in theory.

Also, I'm aware that the tone is different, though very similar. I don't think The Batman's depicition of Gotham, Arkham and society in general are quite as bleak/cynical or the people in it are quite as horrible as in Joker's version, but Joker is seemingly set a long time before The Batman so that could explain the differences there as the society progressed and became a bit less harsh by the time of The Batman. They share a gritty, slightly mature style focused on Gotham's underworld and the grounded nature of only showing criminals and people that could exist in reality.

The one continuity issue I see is regarding what happened to Bruce Wayne's parents, but if that one element was removed could they otherwise be conciliated? Also, couldn't we just argue that the criminals that killed his parents in The Batman were actually Joker's followers just presented differently (and with different actors playing his parents of course)? We didn't see much of that time period when Bruce was a kid in The Batman, so it's possible there was an Arthur Fleck that had started a riot at the time?