Going by our understanding of science and physics as of the time of this posting, how could souls (or something close enough to the idea of the soul to have possibly inspired it) exist and continue consciousness after the death of the physical body, without violating the above-mentioned understanding? As with several phenomena in the real world, like epigenetics or quantum mechanics, the full mechanisms don't have to be completely understood as long as the general concept is.
Expanded conservation of energy.
The atoms making a living body and a dead body are the same.
The only difference between a living body and a dead body (immediately after death) can be accounted on an entity called Soul, which emerges from the superposition of the states of all said atoms.
When a body dies the Soul is not lost, but simply transferred somewhere else.
Philosophically a soul isn't so much a thing as it is a property of a thing, you can't take the "chairness" of a chair from a chair without the chair no longer being a chair, in other words to make a chair not be a chair you need to change it in some way that prevents it being used as a chair.
See: Ship of Theseus
Of course a person isn't just an object like a chair, our identities are not solely defined by our usefulness but also our associations (our interpersonal identity) and our actions/behaviors (our modus operandi and state of mind).
A complete copy of your memories could be considered analogous to a soul however there's no way to determine whether an AI/robot with your memories is authentically you (no doubt psychologically affected by the new state of being) or a new person who just happens to have your memories. Indeed as with the Ship of Theseus the distinction may simply be a matter of perspective, if someone decides that robot is you then to them you are, while someone else may disagree and thus to them you're not, furthermore as the robot you'll have to decide for yourself whether or not we are in fact talking about you.
To answer your question as long as the memories persist you could say someone still exists even if that's only in the memory of other people or stored as some kind of recording. It's a well established trope for people to be haunted by the memory of a lost loved one or someone they've killed, this kind of ghost has no physical presence but is still subjectively real to the person whose mind they inhabit.
Also see: Daemon (novel series)
There are two big theories regarding the universe/universes. The ever-popular infinite paralell universes theory, and the less known but equally intriguing infinite universe theory. The infinite universe theory states that our universe itself is actually infinite. Infinity is a really, really, really big concept. It basically means that over an infinite distance there are an infinite number of probabilities. This means there is an infinite number of every possible event allowed by physics to be ocurring within that universe an infinite number of times at once. The same concept applies to infinite paralell universe theory, but I think its not as neat as being able to say you have infinite copies of yourself within the same universe.
Point being, if this universe truly does extend to infinity beyond the area which we can observe then an individual is simply a cosmic fact. There is never a time within that universe where they do not exist. Every iteration of a person are in a perpetual state of acting out every physically possible scenario infiniteley across an infinite existence.
Now, as far as the soul is concerned its really up for cultural debate what that even is, but if you die somewhere some version of you identical right down to the sub-atomic level is re-living the exact same life, creating the exact same memories, and expieriencing and exhibiting the exact same conscioussness you did. So, if that is the case then one might say that nobody truly ever dies, they simply are a small cosmic fact that transitions its locations and states of being but never really ever leaves the fabric of existence. As far as if that counts as posessing a soul which lives on after the particular iteration of existence that you call yourself ends, well that really is a matter of personal philosophical opinion. I'd like to think that your conscioussness isnt ending once you die, its simply moving its existence somewhere else along the timeline and space within our universe to begin anew.
Even if our universe itself isnt infinite, the same concept applies across an infinite multiverse as well.
At the very least versions of us do continue through DNA to or children. I had dreams not by own and who says its not memories passed through DNA from my ancestors whom continues to exist through me.