Arguably, Batman: Year One, DC Showcase: Catwoman, and Batman: The Dark Knight Returns Parts 1 and 2 were all in the same continuity.
Well, yes, the
Catwoman short was made to be in continuity with
Year One, although it used the modern Darwyn Cooke costume. I'm not sure the continuity relationship between Y1 and TDKR is that clear, though. After all, in the comics, I believe TDKR was meant to be an out-of-continuity conjectural future, or at least it was usually treated that way, while Y1 was the canonical post-Crisis origin story. I gather Miller has claimed that Y1 is real within the continuity of TDKR, but I think TDKR is not necessarily real within the continuity of Y1. And of course the movie adaptations used completely different voice casts.
Also, I've seen some argue that Justice League: Crisis on Two Earths and Justice League: Doom are in the same continuity despite having two different Flashes (Wally for JL:CoTE and Barry for JL

). Personally, I think they're different continuities within an infinite multiverse.
I'm sure there are a number of movies close enough that fans can
believe they're in the same continuity, but I'm talking about what the producers intended, and their official policy prior to 2014 was to make only standalone movies, with rare exceptions.
If this is so, then considering the infinite multiverse shown in JL:CoTE, the DCAU, Batman: The Brave and the Bold universe (Earth-23?), The Batman universe, and the other universes represented at the end of B:TBatB episode "Night of the Batmen" makes even more sense. This is why, considering the similarity of animation styles, I believe them to be part of the same multiverse. I would also include Young Justice (Earth-16), Green Lantern: Emerald Knights universe, Green Lantern: First Flight universe, and perhaps all the other universes of the DC Animated Original Movies line. Maybe a version of COIE could be done with this multiverse.
That seems most unlikely. After all, many of those were made for different target audiences, some were less successful than others (there's a reason the movie line stopped making GL movies), and they had various different producers behind them, so from a professional standpoint (as opposed to a strictly fannish standpoint) there's no percentage in revisiting most of those universes. Not to mention the prohibitive expense of rehiring all those different voice casts.
I also question whether it's desirable. People have this nostalgia for COIE, but while it did have a couple of important, memorable moments like Barry Allen's sacrifice, I think it was an incoherent story, an overcomplicated exercise in trying to eliminate a nonexistent problem that basically just introduced whole new problems and ended up being reversed eventually anyway. The whole thing was a bad idea, and it's had a pretty terrible legacy, because it opened the door to DC rebooting its continuity over and over and just making a ridiculous ongoing mess of the whole thing.
Honestly, fans obsess too much on continuity and "alternate universes" and all that. None of that actually matters, except as an occasional plot device. These are stories. What matters is that they're entertaining. Letting different creators tell stories about the same characters and concepts in different and incompatible ways is entertaining because it allows exploring alternative possibilities and variations on a theme. Any sort of "multiverse" handwaving is usually just a tenuous excuse for those variations, and one that isn't actually necessary for the most part, because we understand that these are just stories and that they're different because different people are telling them. And there's nothing wrong with having those variant stories, and there's absolutely nothing positive to be gained from forcing them all together into a homogeneous mass.
As far as the universe at the end of The Flashpoint Paradox not being the same as the one in JL: War, I'm not so sure. I get what you're saying and ordinarily I'd agree. However, there's still the credits scene at the end of The Flashpoint Paradox which connects the two movies.
Yes, of course that was the vague intent, but it just as obviously didn't work, because the two movies clearly took different approaches. When they put that teaser into TFP, they probably didn't quite know what would happen in the next movie. So it wasn't meant to be a literal "same universe," it was just a narrative hook. One story teasing another is not about defining a shared reality, it's just about making the audience curious enough to come back for the next installment. (Like how third-season
Batman episodes would end with a teaser of the next episode's villain, often directly contradicting the actual events of the next episode. They weren't meant to be in continuity, they were just hooks for the audience.)
The problem with the fannish desire to pretend that stories can be treated as parallel realities is that fiction and reality don't follow the same kind of rules. Stories are much more mutable than reality, and they can be rethought and retconned and patched together and so on, to the point that it's useless even to pretend there's any reality to them. How do you define the "reality" of something like, ohh,
M*A*S*H, an 11-year series about a 3-year war, and its massively self-contradictory internal chronology? Or like
The Six Million Dollar Man, where Steve Austin was a civilian astronaut made bionic by Oliver Spencer in the pilot movie but an Air Force colonel made bionic by Oscar Goldman in the series?
That's why I generally prefer not to assume two different fictional interpretations of a concept represent "parallel universes" unless they're explicitly defined that way within the narrative. Of course, in comics there's a long tradition of pretending all the alternate versions are parallel realities no matter how bizarre and absurd they get, but it's not something I take all that seriously. It isn't important to me to force stories into that paradigm if there's no in-story narrative reason for doing so.