Brandon,
Going back to the comics, John Byrne's, Jeph Loeb's, and Mark Waid's versions of Superman exhibit major personality differences, yet they are all part of "post-Crisis" DC continuity.
Keith Giffen's JUSTICE LEAGUE exhibits a complete tonal difference from Grant Morrison's JLA, as well as
every other DC comic ever published, but it too is within the same continuity.
And Krypto is back in the comics, as well.

Likewise, pre-war, wartime, and post-war Superman can almost be considered 3 separate characters (see the excellent SUPERMAN IN THE FORTIES graphic novel) yet they are all part of the golden age "Earth 2" continuity. The silly 50s Batman, Julius Schwartz's 1964 version, and the TV-era "campy" Batman are likewise all part of the same silver age, "Earth 1" Batman continuity.
Besides, continuity is
overrated. First of all, it exists solely in the mind of the fan -- in any shared universe, dozens of writers and editors are creating stories, and they can't be expected to be familiar with everything that has come before. More importantly, superhero stories are like folk takes -- they change with each retelling. Among my favorite versions of Superman are the Siegel/Shuster original, the Fleischer cartoons, the Richard Donner films, the John Byrne comics, and Mark Waid's new origin story. However, I would never claim that any version is less real or less legitimate because it violates the continuity of the versions that I prefer (none of which are in the same continuity, by the way).
I find it amusing that comic book and superhero fans are so generous with their suspension of disbelief -- we will accept any sci-fi (or pseudo-sci-fi) explanation for anything, whether or not it is actually based on real scientific principles. We will accept any
fantasy explanation for anything, as long as it is more-or-less internally consistent (with a great deal of leeway given in this area). But what we WILL NOT TOLERATE is the most trivial factual inconsistency. Marvel Comics used to parody this mindset by offering a "No Prize" to clever readers who could explain away such inconsistencies in their letter columns.
Myself, I refuse to accept that any Spider-Man stores between Todd McFarlane's departure and J. Michael Straczinsky's arrival ever occurred. That means (to me at least) that there was never any "Obscene Clone Fall" or whatever it was called.