Why would songbird capture our Elizabeth in the 2nd parallel universe when the Elizabeth of that universe was never saved by Booker (he died a martyr before he could reach her)? Was it simply because songbird happen to chance upon her? Also, shouldn't that universe now have two Elizabeths (this wasn't reflected in the game)?
My understanding is that Booker and Elizabeth don't actually travel between the dimensions. Elizabeth creates a superposition of two (or more) universes instead.
This is consistent with how tears work during the combat, how the Siren is brought to (un)life and explains why the guards in the club remember being dead.
"Whenever Elizabeth opens a tear, she is moving to a universe that contains the conditions that she wants or needs at the moment. However, her ability also has a small area of effect, bringing with her people around her in a small radius. We can see this shown when Fink's Head of Security is unaffected, despite being dead in the first timeline. Whenever a tear is opened and the people step through, the mind will attempt to settle paradoxes by rewriting memories.
The first paradox we see is the state of being alive or dead, which affects Chen Lin, Booker, Lady Comstock, and two guards. What happens here is that the memories of the dead and the living are in conflict with each other, so the mind will attempt to combine memories of death with life. The effects of this seem to vary in intensity and results. A common ailment is that people will end up feeling traumatized and deeply disoriented, often seen struggling with their deaths. A possibly more intense reaction is the formation of new abilities, although Lady Comstock's situation is a bit special as 1) Elizabeth's powers were forced by the Siphon and 2) the Lady Comstock brought over is the one that encompasses the traits that Elizabeth imagined her to have .
Booker's situation is a special case as he is actually stepping into another universe. Thus the memories that merge will collect in his mind, which is seen in the case where Booker has memories of leading the Vox revolution. What is interesting is how Booker is unaffected when he's jumping from the first to second timeline in The Good Time Club. My guess is that because Booker is stepping into a timeline where there isn't a Booker. However, because the circumstances would've been the same, the mind does not have to rewrite his memories. This idea is tenuous at best, and could simply be the result of an oversight by the developers unless someone can explain this better.
The second paradox is the state of existing or not existing in the timeline, which is what affects Booker and Robert Lutece. Because Booker steps into a timeline where he doesn't exist, Booker's mind is trying its best to rectify that, muddling his memories to fit within this new universe. Robert Lutece also experiences this as well when he crosses over to meet his sister. This is a type of paradox where the memories of the same people will not merge.
Why is it the second paradox people will merge, but not the first type? Well, I can theorize that it has to do with whether or not the memories merging belongs to the same people. The Chen Lin in the first and second universes have the same identity and are practically similar, so their memories will merge. However, Booker, Comstock, Rosalind, and Robert are largely unaffected by the paradox of being in two places at once, with some minor hemorrhaging on the side of Booker and Robert. This may be due to the fact that there is enough of a difference between them that the universe allows them to exist simultaneously. In the case of the Lutece twins, what separates them is their gender, where combining them would probably induce another paradox of being both male and female. For Booker and Comstock, it is mentioned that Comstock's own identity has been eroded due to repeated exposure to the tears, thus making his identity different to that of Booker's.
So, what about Elizabeth? Why is she not affected in any way? Again, we have a special case where Elizabeth's power over the tears allow her to retain her identity and memory. We cannot say for sure, as the origin of Elizabeth's powers is simply explained with a hand wave of being in two places at once*, and the game doesn't delve deeply into this."
Source: Reddit /r/BioShock post, "Bioshock Infinite: The Ultimate Spoiler FAQ", by Reddit user awchern
Songbird had already captured Elizabeth in both universes. The main difference between the two was whether Booker was successful in the rescue attempt. There wasn't an instance of two Elizabeths in the universe where Booker rescued Elizabeth.