For many decades, the Copenhagen interpretation of Quantum Mechanics has been the most commonly taught. The basic scheme of it as a layman like me understands it, is that quantum mechanics makes objects exist only if they are measured. Einstein was quite opposed to this and asked Niels Bohr, one of the founders, whether he really believed the Moon was there only if he looked at it.
To a degree there is some justification for the idea that objects don't exist unless they are communicating information. For example, the Knowability Paradox states that all knowable truths must be known or they are unknowable. So if someone is in complete isolation from anything in the universe, can we say this person exists in any meaningful or relevant sense? This doesn't preclude the existence of God, angels, or anything outside the universe, because they can convey information to our world should they exist.
But for the purposes of this brief discussion, I'll focus on the implications for Christianity. For example, with the Schrödinger's Cat Experiment, at a certain point according to quantum mechanics, the cat is both dead and alive. One interpretation is that when the wave collapse makes the cat either dead or alive once it's observed, in another universe, the cat is the opposite. This, the Multiworld Interpretation, or MWI, is one explanation of the true randomness of quantum particles, and is a form of superdeterminism. And that's the issue that catches my attention.
The problem of uniqueness is the main issue. If there are an innumerable amount of universes for every possible path of a quantum particle, then how many Jesus Christs are there? The contradiction between the Bible and multiple universes isn't exactly because of this, in my opinion. Obviously a mirror would have his image, so perhaps we could look at these other universes as irrelevant 3D mirrors with different values. However, some of these alternate universes would have version of Jesus where he would be malicious rather than good.
This can perhaps be denied as a Jesus and universe that are only a similacrum of the real ones. If Jesus had a twin, obviously that's a different person. In that case, since MWI is a form of superdeterminism, determinism would have to be accepted for all of those other universes except our own. This may amount to special pleading, but if we posit the existence of free will only for one specific variation, then it's not any more specious than to say that the universe exists with the laws that it does, or that we exist in this current (ever-evolving) variation, since we do. In other words, one cannot say what's probable, let alone possible, with free variables such as other universes. This basic fact is underlined by the admission of supporters of MWI that everything that is possible, no matter how statistically unlikely, would have happened in at least one of those universes.
III. Implications for Other Religions
Honestly, the only other religion besides Christianity that may need to reject MWI to some degree would be Islam, and even that's questionable for reasons that I'll give below. There are bound to be issues with most religious traditions to some degree, especially since they were modeled on a world with a far different philosophy (i.e. what's normally taken for granted: there's only one of me !). But they would have plenty of room to accomodate, as does Christianity perhaps if the above reasoning holds.
One of the main schools of Islam holds that the Qur'an was uncreated, since it's God's Word. In that sense, if MWI was true, then perhaps there would be more than one. But this probably isn't really the case because the Qur'an in Heaven would be one and unique, and only the transmission on Earth would be multiple: just as there are millions of Qur'ans printed today.
Otherwise, perhaps the various angelic appearances could pose an issue, but this would be the same understanding as that for Christianity above: other universes merely have particle-based similacrums.
Judaism has even fewer problems. The Saducean sect in Jesus' day and before didn't even acknowledge the existence of any metaphysical soul, so a multiverse would have very few problems. The fact that the number of sins would be vastly higher would be negated by the fact that the number of repentant and good works would be too.
Angels that appear in the Hebrew Bible, would have to be similacrums, but is that really a problem? Aren't their physical representations not their true form anyway? In Judges 13:5 the angel reveals to Samuel's parents that his name can't be pronounced by any sound on Earth, so how much more his appearance. Hence why the Israelites died when looking into the Ark of the Covenant (1 Sam. 6:19), and how Moses could see God's back (Exodus 33:19-23).
All the different incarnations, or avatars, of the various popular deities in Hinduism would if anything welcome the Multiple-World Interpretation. The reincarnation would probably integrate easily too. Even though both of these are different, it would probably make more sense intuitively than the Abrahamic religions. The one potent objection to reincarnation: the increase in humans despite the increased human potential for destruction (which would mean more humans should be reincarnated as lower animals), could be answerable even better under MWI, even though statistically speaking it probably shouldn't (as it contains all variations, so there's no actual change in karma!).
Buddhism and Jainism
The mainland Indian tradition of Buddhism, influenced quite a bit by Hinduism, would integrate MWI easily as well. The different vehicles for spiritual energy could easily be seen as coming from other realities. This would easily be seen as the source of the Dalai Lama or the Jain Tirthankaras. Enlightenment and Nirvana may or may not relate to MWI.
Original Buddhism would probably be unaffected as it relies on a very Earthly philosophy of wisdom, temperance, prudence, etc. Much like Greek philosophy, it's simply a tool for better life in the natural world.
The numerous local religions throughout the world would have little difficulty in absorbing the idea of MWI. In particular, the Aztec cycles could be something that even seems to predict/propose the idea.
The testability of MWI is technically possible but beyond current capabilities. [The Wiki on Many-Worlds Interpretation] The majority of physicists seem to not accept MWI mainly because it posits more unverifiable unknowns than necessary: it's untestable and therefore somewhat unnecessary.
On the other hand, those who accept it seem reticent to concede the randomness and other aspects of Quantum Mechanics such as action at a distance. My personal feelings are that, even if we assumed an atheistic universe, it's easier to explain things without MWI because either way you have random properties (e.g. the laws of physics being how they are) with no origin or plan. Quantum entanglement therefore doesn't need to be anything different from how the Higgs boson generates mass in other particles, or why there is a Baryon asymmetry.
There's probably a lot of inaccuracies above. But this is a tantalizing question about the nature of reality. Those with this kind of curiosity are lucky we live in these times with all this knowledge. One can only be reminded of what Jesus said about the older prophets and believers regarding his arrival (Matt. 13:17).