Okay, I feel I should open this with the disclaimer that I've not played the Pre-Sequel or Tales from the Borderlands, so if there is information pertaining to the nature of Sirens in either of these sources that contradicts what I have here, I sincerely apologise, I'm just working on what I have as of the Borderlands 2 DLC at the latest. Now then, without further ado, let's jump right in with the facts.
P.S. There will be spoilers.
1. Sirens have powers other humans don't.
As is obvious, each Siren has shown to possess a power no other human has, or other Siren for that matter. Lilith's Phase Walk, Maya's Phase Lock and Angel's Phase Shift, all alter forces in ways normal humans can't.
2. Sirens have naturally occuring tattoos
An odd trait, but each siren has a unique pattern of tattoos running down one side of their body, seemingly from head to toe.
3. Sirens are born, not made.
Evidence of this is shown in the three primary sirens in the series, we know Lilith has had her tattoos since childhood, Maya was raised as the messiah of some insane Order and Angel is Handsome Jack's biological daughter.
4. Sirens have the ability to absorb Eridium more easily than other lifeforms.
As we've seen with Bloodwing and other mutant beasts across Pandora, exposure to raw eridium isn't exactly healthy, for Sirens however, it can actually serve to enhance their powers. The downside is that it is more like a drug to them, and they become increasingly more dependant until going without it proves fatal. This can be seen in Lilith's use of the substance in 2, along with Angel's ultimate fate.
5. Sirens, through accident or design, are connected to the Vaults
With their ability to charge the vault keys and powers that set them apart from other humans, it's clear the sirens are connected to these ancient artefacts, whether this was intentionally done (most likely given the precise numbering of Sirens) or some fault of evolution and exposure is unclear.
6. There are only Six Sirens
As stated by Handsome Jack, there are only Six Sirens in the universe. Whether this means at any given time, or that only six have been found across the known universe, I have no idea, but there's only six.
These are things we can assume are true based on evidence so far.
1. All Sirens are women.
The name alone harkens back to the female creatures of myth who lured sailors to their dooms and of the four Sirens shown in the series thus far, not one has been male, and with such a small group to pull from, the odds of there being a male Siren seem pretty small at this point.
2. Each Siren's power is unique to them.
As we've already addressed, the Sirens have inhuman abilities that are seemingly not shared. While they do have similar traits, all three main Sirens being shown to manifest wings made of elemental energy, absorb Eridium etc. their unique 'Phase' abilities don't seem to be shared.
3. Siren powers are not inherited directly through genetics.
Given that it is stated by Handsome Jack that there are only ever six Sirens at a time, it's safe to assume that, even if it is an inherited trait, it is from somewhere far back in Human History as far as Borderlands goes and is likely a recessive trait, only emerging when necessary. The evidence here being that none of the Sirens have shown a direct familial relationship with one another, and if the trait was passed down directly say from a mother to a daughter, then that would create a potential twelve Sirens at once, which is an apparent impossibility. Now I'm sure someone will likely bring up that of the Sirens we've seen, we don't even know their mothers, it could be one dies when the other is born or something, but a single audio log from Handsome Jack helps to prove this is not the case. In one Echo, Jack is forcing his daughter to coply with his demands and she asks where her mommy is. This implys that it would be a normal thing for her mother to be there, so if it was a case that Siren powers follow some sort of direct bloodline, something would have to have gone very wrong for Angel's mother to survive through her daughter's formative years.
4. Sirens are valuable
Not so much an inference as a straight up fact, but value is subjective, so I'll put it here in order to illustrate that. Since there are only Six Sirens, like any commodity, that makes them exceedingly rare and valuable to people. Evidence of this is clear, Commandant Steele achieved a high ranking position in Atlas' private army. Maya was used as the weapon of an insane Religious Order to keep an entire planet in line. Even Angel, the daughter of a low level programmer at Hyperion, was, before Jack even took control of the Company, given control of monitoring functions over the Echonet and a orbiting surveilance satellite. After their relation to the vault became apparent, their ability to drastically reduce the time needed to charge the vault key and harness it makes them even more useful to people with that interest. The only note of someone not valuing Sirens, is the Vault Creatures themselves. The Destroyer killed one, and both it and the Warrior show absolutely no hesitation in attacking others. This alone wouldn't be weird, except that the Destroyer was an Eridian Prisoner, and the Warrior, their Weapon. Two sides, and yet neither had hesitation of a failsafe or anything, preventing them from slaying those with powers. I know, game mechanics and all that, but it seems the Eridians would make it so their own weapons don't attack them, so just how close the Sirens are to these ancient aliens is called into question when bother their enemy and their pet are willing to squash them without hesitation.
The following are some of my own thoughts towards the nature and possible existence of Sirens.
1. There are more Sirens out there
Pretty obvious one, but I'll present my evidence. While Jack was very adamant about the number of Sirens, I'd like to start by pointing out that, as the Hitchhikers' Guide to the Galaxy states "Space is big. Really, really, really big." So the possibility that in some distant corner there's another Siren or two, wouldn't surprise me. On top of that, let's be blunt about it, Jack is, not the sharpest tool in the shed. Certainly cunning and manipulative, but he is the man who owns a diamond pony named 'Butt Stallion' who thinks words like Ergonomic and Perpendicular are classy and make excellent names for firearms, so it wouldn't surprise me if he made a mistake.
2. There are at least two prepubesant Sirens
Again, fairly obvious, but if the number of Sirens is constant throughout the universe, then when Steele and Angel died, two more must have been born to take their place, or at least concieved. So as of the end of Borderlands 2, there must be a five year old Siren to replace Steele, and one anywhere between just concieved and just born to replace Angel.
3. Krieg was married to a Siren
Yeah, I know, this one comes out of left field, but I have evidence. Lots and lots of very circumstantial evidence, which is why it's in personal theories. First off, Bandits and Psychos aren't that bright normally. So while Jack might know there's only six of these people in the known universe, the bandits and things thought that 1. Lilith was a God 2. They could pass off women as Sirens by copying some tattoos onto them and 3. That they could take such a person in a fight. All, very, very dumb ideas.
Krieg on the other hand, the instant Maya stepped off the train, identified her gun, her occupation and the fact that she's a Siren within seconds. Now I know it was the rational part of his brain doing it, but everything his rational brain thinks, first has to be absorbed by the conductor of the poop train. So in a matter of seconds, this Psycho, a man whose mind is fractured, puts together information that scores of paid bandits, assassins and psychos couldn't. He assesses the situation, dives for cover and pretty much surmises that he's dead trying to fight a Siren while he otherwise gleefully fights anything.
It's also during this exchange that he actually starts to regain parts of his humanity. He pulls it together enough to warn Maya and save her from a Rat and he also wants her to save him. That's a hell of a fixation to form in a matter of minutes, especially for the rational part of Krieg's psyche which shouldn't be as prone to obsessive behaviours. Why would he do this? Possibly because he associates, somewhere in his busted up brain, the Siren, with love and healing, so what could have previously been feelings for someone else in the past, could be transferred to someone like that.
Now then, there is a lot of evidence in text and things that I'll skim through that basically paint Krieg's life before now. Popular opinion being he was a vault hunter/adventurer himself, possibly married with a kid, before being snapped up for Hyperion experimentation. Went mad, escaped, and has been a wandering psycho ever since. Now assuming these facts are true, this lends credence to the idea that sometime in his travels, he could have met and formed a relationship with one of the unaccounted for Sirens, but the most damning of evidence for me, comes from the Mad Moxxxi and the Wedding Day Massacre DLC.
During a quest, Moxxi will ask the player if there was ever a Mister or Missus Vault Hunter. When Krieg is asked this, he replies with
"The tiny one and the sad faced mother and the blue tattoo forever and ever."
Or words to that close approximation, I may have misphrased it, but the bones of it are all in there. A 'tiny one' likely a child, course with Krieg, a small car could also count. A 'sad faced mother' likely Krieg's wife and the blue tattoo, which is an odd trait for his mind to latch onto unless it was very significant, like it would be for a Siren.
Again, this is just a theory and it is based almost entirely on circumstantial evidence, the only hard facts being that Krieg is crazy as balls and he's fixated on a Siren now.
Anyway, that about wraps it up for now. I plan to write a new one in the coming months addressing theories and speculations on the vaults and their purposes. In a final note, I'm rather disappointed there's no Playable Siren in the Pre-Sequel, they missed their shot at the trifecta.