Like the title states, how would it be possible for the same species to live on multiple planets (before space travel is invented)? Basically, a civilization invents space travel, meets aliens, and surprise, they look almost the same. (common occurrence on sci-fi shows). There can (and should) be some difference in appearance, somewhat like different races on Earth, or dwarfs vs elves kind of deal, but they should be genetically compatible to breed.
The only way this can happen is if there is a link between the two planets. The easiest way to explain that is an alien transplanting a group from one planet to the other for whatever reason.
The link would need to be relatively recent too, if you transplanted Homo Erectus for example then you might get the "dwarves vs elves" scenario. If you transplanted something further back though then the odds of them being even remotely human are pretty slim.
Since evolution is random, the chance that any two species evolving very similarly without being connected is close to zero. But not exactly zero, because the fact that there is one race of beings proves this route worked. Therefor in an infinitely large universe it is even very likely that there is some place somewhere where a similar looking race evolved. Chances they meet, though, should be very rare, unless you find a way of travelling really, really fast. In SciFi movies the main reason for aliens looking similar to us is that you need actors, and they are of the same species. The second reason is that people tend to think of sapient creatures as having the form of people, because that is all we know.
You could play with parallel universes.
You ask about having the same species originating from different planets: could it be ok if it was the same species originating on the same planet, only in different universes? I think it is less boring than alien intervention, and more "possible" than parallel evolution.
One day, humans break the barrier of light speed and discover the possibility to move between different parallel universes and meet different versions of our own Earth. In many of these alternative Earths, Humans have evolved in very similar ways as ours, but most of the historical events were different, making their present day totally different. Here, the Roman Empire never fell; there, World War III led to a new Middle Ages; there, Nehanderthal won over Sapiens.
as already stated it is extremely unlikely, astronomically so, that two identical species will evolve on independent worlds.
If you desire that to happen the most plausible explanation is to use the 'seeder' approach. Some powerful entity or entities seeded similar life on the two planets to jump-start evolution towards the same end goal. Even then they would have had to keep playing a role in evolution until a very recent time (from an evolutionary standpoint, where thousands of years is an eye blink) for the two to have evolved similarly enough to be thought of as identical or near-identical species.
This is, of course, a very unlikely scenario. However, it's still more plausible then random evolution leading to near identical species both evolving and meeting each other. Plus, as a writer you have plenty of freedom to make up details like this in your backstory. You can always kill the species off in your backstory if you don't want them existing to complicate the plot in 'present day' of the story.
The simplest way of putting it is that the species from the different planets can only look similar if the planets also look similar. What I mean is that if the planets have similar environments, ecosystems, weather and/or what-not, the chance of similar beings popping up on them becomes very plausible.
I'm going to make some assumptions about the question and why you're asking it. You're either: wanting romance between aliens and humans, and you want there to be children from their union; or wanting sex between them, and want impregnation possible from their union. Based on the question, I'm leaning towards the former.
So "same species" in this context just means "close enough that viable hybrids can be formed".
And I'm going to assume that by having an origin on different planets, you mean that they both inhabit different planets, with lengthy histories, and no reason, even in the face of science, to believe their origin anything other than being from the planet.
So let's look at the ways this has been dealt with before, then see if we can think up any more.
One way this is very commonly dealt with in fiction is the "they were transplanted back in caveman days and didn't know it". For earth-humans, this just Does Not Work. From a narrative point of view, it is more "interesting" to have the earth-humans be the transplanted ones, because then you can pull the big reveal at the end. Except... nowadays, this just becomes a giant facepalm. It is impossible to write a story that fits with all the pelaeoanthropological and DNA evidence as we have it today, and yet has humans transplanted in. Don't go there.
Having the other guys be humans transplanted from Earth many centuries ago could work, but feels like a cheap hand-wave.
Those two are the only "actually the same species" scenarios I can think of. You cannot have one species with two different origins.
Then there's panspermia and convergent evolution, which has the disadvantage that anyone who knows the topic will just laugh and facepalm. No matter how visually similar two groups are, their DNA is not going to be anything similar to a match.
Panspermia does have the advantage that it means you don't need a third party to "seed" the planets, and it explains why both planets use DNA to store their life-blueprints. So it's at least a good background to other ideas.
Convergent evolution is okay if you don't look too closely. Cases of convergent evolution tend to be caused by a very specific niche or need, starting out with similar building blocks. The human form just doesn't fit that: our form is the result of too many weird accidents. In the entirety of mammalian history, no, the entire history of vertebrates... no, in the entire tree of life on our planet, there's only been one tiny branch -- one twig, really -- containing large-brained bipeds. If you also make them naked, taill-less, lactating, with vocal chords and hair on their heads... you get the point. You can handwave with "quadrupedality is the most sensible form; bipedality and large brain size are natural results of tool use" and most people will accept it without too many qualms, so long as your aliens look significantly "different" and "alien".
So, given "having identical DNA" is out, let's look at what's needed for two dissimilar species to produce a viable hybrid. This is an area I know little about, but basically, for a sperm and egg to form a complete cell, they need very-closely-compatible DNA. The likelihood of two completely different species reproducing should be about zero: that's sort of the definition of the term "species"... but that the term "hybrid" exists shows that some species can interbreed. Generally only very closely related ones, though.
Then there's "locally built autonomous avatars". In Species, scientists build a creature from human DNA using modifications sent from the alien civilization, so that the creature was programmed to behave as a representative of the alien species. Doesn't work out well in the movie, but the principle kinda-sorta works, so long as all you want for your representative on Earth is a set of very basic behavioral instincts, in a human upbringing. This is kind of a nice solution since it gets around the speed-of-light travel problem.
If there's a way to transmit the brain as data across the vastness of space, and then to imprint a mind onto a "blank" or "host" brain, that again gets across the travel problem. This has usually been written about where people in insane asylums or children are the most "receptive" to being hosts, and historical cases of "demon possession" are used to support the story.
Then there's parasites. Ridley Scott's Aliens series does this well: the aliens grow in the host "mother", apparently stealing some of the host's DNA, or at least co-opting the growth process so that the result has a similar shape and size as the host, but introducing an alien mind and some alien physiology.
Then there's deliberate body modification by more advanced aliens. If they're advanced enough for interstellar travel, they're advanced enough to think little of the necessary plastic surgery to modify their bodies to suit local climes. This would explain any extreme similarity to humankind, and in an advanced enough alien society, could also explain genetic compatibility ("yes, dear: we modified ALL our bodies to be compatible with yours: it's not just skin deep. My culture sees romance and family building as an important part of diplomatic relations.")
Though there's also skin-deep rubber-mask fakery, of course, or trivial plastic surgery, but that still requires bipedality, and still doesn't make breeding possible.
But for that, there's science-assisted pregnancy. Never mind the natural method, fun though it might be. Alien and lover go into the flying saucer, get strapped to a table, and get impregnated with the very latest in scientifically engineered zygotes, designed to represent the best of both parties, and to be capable of using the mother's normal reproductive system for growth and life support. This is a handwave, in that the science isn't explained, but it gets rid of the incompatibility problem by saying that a computer worked out the incompatibilities. It's still a little difficult, the baby would have to be a lot closer in genetic form to the mother than the father.
For equal portions of both, could also just gestate it in a test tube the whole way. This is arguably not as good as having a real mother, but would work where there's a perceived danger from incompatibility between mother and baby. Downside is that the mother normally grants a whole bunch of immunities and suchlike that a test tube would not, so might be a sickly baby. And also a real earth mother makes for a better story.
Simple: declare it thus
Surprisingly there is not an objective way to declare whether two individuals are of the same species or not. There's some general rules of thumb (if they can't mate, they must be of different species), but the real answer is blurrier than you'd think. Consider bacteria, which often share genetic information with other bacteria well outside their species.
So if you wanted to declare us the same species, you could.
Now, what you're probably looking for is two individuals from different planets which seem sufficiently similar that the average young adult would not choose to call them different species. That is more interesting. The odds of this happening are unimaginably huge. If you rounded down to zero, you might not be wrong. I wouldn't worry about it in terms of 1: 10000.... with some number of zeros, the number of zeros would be so big that I'd want to talk about how many zeros the number of zeros has!
So if you're going to go for this, go for it with glory. Hit the problem square in the nose and ask for forgiveness later. Use convergent evolution -- the idea that there is a structure of a body which is particularly suited for living, and all humanoids approach it.
I think it smacks of egotism to say we are actually the universe's ultimate creation, but for a storyline, it can work. The trick is that you wont just find one or two planets converging in this way. Expect to find humanoids on hundreds, thousands, even millions of planets scattered across the galaxy. Expect humanoids to be so common that we get frustrated with how similar everything is.
Characters should feel like there is some cosmic thread that they just can't quite tug on, but its always there. Their frustration will help make the reader feel more comfortable with the unbelievable handwaving. At least the characters agree that there is more than meets the eye.
I am not sure if this is what you are after but, here goes:
While "same species" would be scientifically impossible, "very similar looking" species is not so far fetched.
Suppose you have two water-covered worlds. It is feasible to have shark-like predators in both. Convergent evolution would agree with it. If conditions are similar in both worlds (temperatures, water depth and so on) two very similar species might evolve and be externally indistinguishable. Highly unlikely, but no impossible: sharks, dolphins and killer whales share the same shape but are very distinct.
However, if you want the same species, is easier to go for the transplanted individuals option.
Warning : The following paragraph is tongue-in-cheek sarcasm, meant to entertain the reader...
Not to set on the sacred robes of the Holy Darwinists, but evolution is still just a theory. It is a good one, but not necessarily true in all of its conclusions. Especially when you expand its scope to include aliens.
This is were the actual answer begins...
For example, @Tim B asserted that the alien transplanting link must have occurred fairly recently because at the speed of evolution, our space cousins wouldn't be even remotely human if they predated Homo Erectus's arrival at c.1.9 million years ago. This assertion about the speed of evolution appear valid when the Earth is an isolated and closed biological system; but looses credibility when alien interaction comes into play.
For all we know, the bone scraps and skulls that we have attributed to Homo Erectus might actually be the remains of the forerunner aliens who seeded us. Or the H.E. may just be the latest version of life which those alien's deposited here at that time. To that end, the semi-unbroken line of DNA tracing back through our planet's history might not be proof of evolution. Instead it might just be the signature of the single Vitaformer* who has been responsible for all the generations of life which were deposited here.
Vitaforming - a sub-specialty of terraforming which focuses on the creation of a balanced, self-sustaining and living ecology within the planetary re-purposing process.
I'm not throwing out evolution... only realizing that with the possibility of alien manipulation, much of our assumptions about evolution's rate and cause may be incorrect.
Higher life form evolution might actually run significantly slower than we have assumed. It might take hundreds of millions of years to splinter into separate species. Which would make alien seeding of multiple planets much more understandable, even when considering the light speed barrier. In this slower evolving universe, we can remain aesthetically identical and reproductively compatible with our space cousins long enough to create the technology to reach them, even if our adopted planets are separated by hundreds of light years.
I imagine a great Forerunner farmer, plowing his galactic field, planting a crop of Homo Sapiens along this spiral arm, just far enough apart to keep them from interfering with each other during their early growth.
Beware... Our harvest time may be approaching.